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<head><title>Ring -- the class of all rings</title>
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<div><h1>Ring -- the class of all rings</h1>
<div class="single"><h2>Description</h2>
<div>Common ways to make a ring:<ul><li><span><a href="___Ring_sp_sl_sp__Ideal.html" title="make a quotient ring">Ring / Ideal</a> -- make a quotient ring</span></li>
<li><span><a href="___Ring_sp__Array.html" title="the standard way to make a polynomial ring">Ring Array</a> -- the standard way to make a polynomial ring</span></li>
<li><span><a href="___G__F.html" title="make a finite field">GF</a> -- make a finite field</span></li>
</ul>
Common functions for accessing the variables or elements in a ring:<ul><li><span><a href="_use_lp__Ring_rp.html" title="install ring variables and ring operations">use(Ring)</a> -- install ring variables and ring operations</span></li>
<li><span><a href="_generators_lp__Ring_rp.html" title="the list of generators of a ring">generators(Ring)</a> -- the list of generators of a ring</span></li>
<li><span><a href="_numgens_lp__Ring_rp.html" title="number of generators of a polynomial ring">numgens(Ring)</a> -- number of generators of a polynomial ring</span></li>
<li><span><a href="___Ring_sp_us_sp__Z__Z.html" title="get a ring variable by index">Ring _ ZZ</a> -- get a ring variable by index</span></li>
</ul>
Common ways to get information about a ring:<ul><li><span><a href="_char.html" title="computes the characteristic of the ring or field">char(Ring)</a> -- computes the characteristic of the ring or field</span></li>
<li><span><a href="_coefficient__Ring.html" title="get the coefficient ring">coefficientRing(Ring)</a> -- get the coefficient ring</span></li>
<li><span><a href="_dim_lp__Ring_rp.html" title="compute the Krull dimension">dim(Ring)</a> -- compute the Krull dimension</span></li>
</ul>
Common ways to use a ring:<ul><li><span><a href="___Ring_sp^_sp__Z__Z.html" title="make a free module">Ring ^ ZZ</a> -- make a free module</span></li>
<li><span><a href="___Ring_sp^_sp__List.html" title="make a free module">Ring ^ List</a> -- make a free module</span></li>
<li><span><a href="_vars_lp__Ring_rp.html" title="row matrix of the variables">vars(Ring)</a> -- row matrix of the variables</span></li>
</ul>
</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_rings.html" title="">rings</a></span></li>
</ul>
</div>
<div class="waystouse"><h2>Types of ring :</h2>
<ul><li><span><a href="___Engine__Ring.html" title="the class of rings handled by the engine">EngineRing</a> -- the class of rings handled by the engine</span></li>
</ul>
<h2>Functions and methods returning a ring :</h2>
<ul><li><span><a href="_ambient_lp__Galois__Field_rp.html" title="corresponding quotient ring">ambient(GaloisField)</a> -- corresponding quotient ring</span></li>
<li><span>ambient(QuotientRing), see <span><a href="_ambient_lp__Ring_rp.html" title="ambient polynomial ring">ambient(Ring)</a> -- ambient polynomial ring</span></span></li>
<li><span><a href="_ambient_lp__Ring_rp.html" title="ambient polynomial ring">ambient(Ring)</a> -- ambient polynomial ring</span></li>
<li><span><a href="../../ReesAlgebra/html/_associated__Graded__Ring.html" title="the associated graded ring of an ideal">associatedGradedRing</a> -- the associated graded ring of an ideal</span></li>
<li><span>coefficientRing(Ring), see <span><a href="_coefficient__Ring.html" title="get the coefficient ring">coefficientRing</a> -- get the coefficient ring</span></span></li>
<li><span><a href="../../IntegralClosure/html/_integral__Closure_lp__Ring_rp.html" title="compute the integral closure (normalization) of an affine domain">integralClosure(Ring)</a> -- compute the integral closure (normalization) of an affine domain</span></li>
<li><span><a href="_minimal__Presentation_lp__Ring_rp.html" title="compute a minimal presentation of a quotient ring">minimalPresentation(Ring)</a> -- compute a minimal presentation of a quotient ring</span></li>
<li><span>prune(Ring), see <span><a href="_minimal__Presentation_lp__Ring_rp.html" title="compute a minimal presentation of a quotient ring">minimalPresentation(Ring)</a> -- compute a minimal presentation of a quotient ring</span></span></li>
<li><span><a href="_new__Ring.html" title="make a copy of a ring, with some features changed">newRing</a> -- make a copy of a ring, with some features changed</span></li>
<li><span><a href="../../ReesAlgebra/html/_normal__Cone.html" title="the normal cone of a subscheme">normalCone</a> -- the normal cone of a subscheme</span></li>
<li><span><a href="../../ReesAlgebra/html/_rees__Algebra.html" title="compute the defining ideal of the Rees Algebra">reesAlgebra</a> -- compute the defining ideal of the Rees Algebra</span></li>
<li><span><a href="_ring.html" title="get the associated ring of an object">ring</a> -- get the associated ring of an object</span></li>
<li><span><a href="___Ring_sp_st_st_sp__Ring.html" title="tensor product">Ring ** Ring</a> -- tensor product</span></li>
<li><span>coefficientRing(SchurRing), see <span><a href="../../SchurRings/html/___Schur__Ring.html" title="the class of all Schur rings">SchurRing</a> -- the class of all Schur rings</span></span></li>
<li><span><a href="../../ReesAlgebra/html/_special__Fiber.html" title="special fiber of a blowup">specialFiber</a> -- special fiber of a blowup</span></li>
<li><span>symmetricAlgebra(Module), see <span><a href="_symmetric__Algebra.html" title="the symmetric algebra of a module">symmetricAlgebra</a> -- the symmetric algebra of a module</span></span></li>
<li><span><a href="_tensor_lp__Ring_cm__Ring_rp.html" title="tensor product">tensor(Ring,Ring)</a> -- tensor product</span></li>
<li><span><a href="_trim_lp__Ring_rp.html" title="">trim(Ring)</a></span></li>
</ul>
<h2>Methods that use a ring :</h2>
<ul><li><span>Ideal * Ring, see <span><a href="__st.html" title="a binary operator, usually used for multiplication">*</a> -- a binary operator, usually used for multiplication</span></span></li>
<li><span>MonomialIdeal * Ring, see <span><a href="__st.html" title="a binary operator, usually used for multiplication">*</a> -- a binary operator, usually used for multiplication</span></span></li>
<li><span>Ring * Ideal, see <span><a href="__st.html" title="a binary operator, usually used for multiplication">*</a> -- a binary operator, usually used for multiplication</span></span></li>
<li><span>Ring * MonomialIdeal, see <span><a href="__st.html" title="a binary operator, usually used for multiplication">*</a> -- a binary operator, usually used for multiplication</span></span></li>
<li><span>Ring * RingElement, see <span><a href="__st.html" title="a binary operator, usually used for multiplication">*</a> -- a binary operator, usually used for multiplication</span></span></li>
<li><span>Ring * Vector, see <span><a href="__st.html" title="a binary operator, usually used for multiplication">*</a> -- a binary operator, usually used for multiplication</span></span></li>
<li><span>Ideal == Ring, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>MonomialIdeal == Ring, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>Ring == Ideal, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>Ring == MonomialIdeal, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>Ring == ZZ, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span>ZZ == Ring, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span><a href="___Affine__Variety_sp_st_st_sp__Ring.html" title="a binary operator, usually used for tensor product or Cartesian product">AffineVariety ** Ring</a> -- a binary operator, usually used for tensor product or Cartesian product</span></li>
<li><span>basis(InfiniteNumber,InfiniteNumber,Ring), see <span><a href="_basis.html" title="basis of all or part of a module or ring">basis</a> -- basis of all or part of a module or ring</span></span></li>
<li><span>basis(InfiniteNumber,List,Ring), see <span><a href="_basis.html" title="basis of all or part of a module or ring">basis</a> -- basis of all or part of a module or ring</span></span></li>
<li><span>basis(InfiniteNumber,ZZ,Ring), see <span><a href="_basis.html" title="basis of all or part of a module or ring">basis</a> -- basis of all or part of a module or ring</span></span></li>
<li><span>basis(List,InfiniteNumber,Ring), see <span><a href="_basis.html" title="basis of all or part of a module or ring">basis</a> -- basis of all or part of a module or ring</span></span></li>
<li><span>basis(List,List,Ring), see <span><a href="_basis.html" title="basis of all or part of a module or ring">basis</a> -- basis of all or part of a module or ring</span></span></li>
<li><span>basis(List,Ring), see <span><a href="_basis.html" title="basis of all or part of a module or ring">basis</a> -- basis of all or part of a module or ring</span></span></li>
<li><span>basis(List,ZZ,Ring), see <span><a href="_basis.html" title="basis of all or part of a module or ring">basis</a> -- basis of all or part of a module or ring</span></span></li>
<li><span>basis(Ring), see <span><a href="_basis.html" title="basis of all or part of a module or ring">basis</a> -- basis of all or part of a module or ring</span></span></li>
<li><span>basis(ZZ,InfiniteNumber,Ring), see <span><a href="_basis.html" title="basis of all or part of a module or ring">basis</a> -- basis of all or part of a module or ring</span></span></li>
<li><span>basis(ZZ,List,Ring), see <span><a href="_basis.html" title="basis of all or part of a module or ring">basis</a> -- basis of all or part of a module or ring</span></span></li>
<li><span>basis(ZZ,Ring), see <span><a href="_basis.html" title="basis of all or part of a module or ring">basis</a> -- basis of all or part of a module or ring</span></span></li>
<li><span>basis(ZZ,ZZ,Ring), see <span><a href="_basis.html" title="basis of all or part of a module or ring">basis</a> -- basis of all or part of a module or ring</span></span></li>
<li><span><a href="___Chain__Complex_sp_st_st_sp__Ring.html" title="a binary operator, usually used for tensor product or Cartesian product">ChainComplex ** Ring</a> -- a binary operator, usually used for tensor product or Cartesian product</span></li>
<li><span><a href="_chain__Complex_lp__Ring_rp.html" title="make an empty chain complex over a ring">chainComplex(Ring)</a> -- make an empty chain complex over a ring</span></li>
<li><span>char(Ring), see <span><a href="_char.html" title="computes the characteristic of the ring or field">char</a> -- computes the characteristic of the ring or field</span></span></li>
<li><span>conductor(Ring), see <span><a href="../../IntegralClosure/html/_conductor.html" title="the conductor of a finite ring map">conductor</a> -- the conductor of a finite ring map</span></span></li>
<li><span><a href="_degree_lp__Ring_rp.html" title="">degree(Ring)</a></span></li>
<li><span>degreeLength(Ring), see <span><a href="_degree__Length.html" title="the number of degrees">degreeLength</a> -- the number of degrees</span></span></li>
<li><span><a href="_degrees_lp__Ring_rp.html" title="degrees of generators">degrees(Ring)</a> -- degrees of generators</span></li>
<li><span><a href="_degrees__Ring_lp__Ring_rp.html" title="the ring of degrees">degreesRing(Ring)</a> -- the ring of degrees</span></li>
<li><span>diagonalMatrix(Ring,List), see <span><a href="_diagonal__Matrix_lp__Ring_cm__Z__Z_cm__Z__Z_cm__List_rp.html" title="make a diagonal matrix from a list">diagonalMatrix(Ring,ZZ,ZZ,List)</a> -- make a diagonal matrix from a list</span></span></li>
<li><span><a href="_diagonal__Matrix_lp__Ring_cm__Z__Z_cm__Z__Z_cm__List_rp.html" title="make a diagonal matrix from a list">diagonalMatrix(Ring,ZZ,ZZ,List)</a> -- make a diagonal matrix from a list</span></li>
<li><span><a href="_dim_lp__Ring_rp.html" title="compute the Krull dimension">dim(Ring)</a> -- compute the Krull dimension</span></li>
<li><span><a href="_euler_lp__Ring_rp.html" title="Euler characteristic">euler(Ring)</a> -- Euler characteristic</span></li>
<li><span><a href="_eulers_lp__Ring_rp.html" title="list the sectional Euler characteristics">eulers(Ring)</a> -- list the sectional Euler characteristics</span></li>
<li><span>Ext(Ideal,Ring), see <span><a href="___Ext_lp__Module_cm__Module_rp.html" title="total Ext module">Ext(Module,Module)</a> -- total Ext module</span></span></li>
<li><span>Ext(Module,Ring), see <span><a href="___Ext_lp__Module_cm__Module_rp.html" title="total Ext module">Ext(Module,Module)</a> -- total Ext module</span></span></li>
<li><span>Ext^ZZ(Matrix,Ring), see <span><a href="___Ext^__Z__Z_lp__Matrix_cm__Module_rp.html" title="map between Ext modules">Ext^ZZ(Matrix,Module)</a> -- map between Ext modules</span></span></li>
<li><span>Ext^ZZ(Ideal,Ring), see <span><a href="___Ext^__Z__Z_lp__Module_cm__Module_rp.html" title="Ext module">Ext^ZZ(Module,Module)</a> -- Ext module</span></span></li>
<li><span>Ext^ZZ(Module,Ring), see <span><a href="___Ext^__Z__Z_lp__Module_cm__Module_rp.html" title="Ext module">Ext^ZZ(Module,Module)</a> -- Ext module</span></span></li>
<li><span><a href="___Fano_lp__Z__Z_cm__Ideal_cm__Ring_rp.html" title="Fano scheme">Fano(ZZ,Ideal,Ring)</a> -- Fano scheme</span></li>
<li><span>flattenRing(Ring), see <span><a href="_flatten__Ring.html" title="write a ring as a (quotient) of a polynomial ring over ZZ or a prime field">flattenRing</a> -- write a ring as a (quotient) of a polynomial ring over ZZ or a prime field</span></span></li>
<li><span>frac(Ring), see <span><a href="_frac.html" title="construct a fraction field">frac</a> -- construct a fraction field</span></span></li>
<li><span><a href="_genera_lp__Ring_rp.html" title="list of the successive linear sectional arithmetic genera">genera(Ring)</a> -- list of the successive linear sectional arithmetic genera</span></li>
<li><span><a href="_generators_lp__Ring_rp.html" title="the list of generators of a ring">generators(Ring)</a> -- the list of generators of a ring</span></li>
<li><span>genericMatrix(Ring,RingElement,ZZ,ZZ), see <span><a href="_generic__Matrix.html" title="make a generic matrix of variables">genericMatrix</a> -- make a generic matrix of variables</span></span></li>
<li><span>genericMatrix(Ring,ZZ,ZZ), see <span><a href="_generic__Matrix.html" title="make a generic matrix of variables">genericMatrix</a> -- make a generic matrix of variables</span></span></li>
<li><span>genericSkewMatrix(Ring,RingElement,ZZ), see <span><a href="_generic__Skew__Matrix.html" title="make a generic skew symmetric matrix of variables">genericSkewMatrix</a> -- make a generic skew symmetric matrix of variables</span></span></li>
<li><span>genericSkewMatrix(Ring,ZZ), see <span><a href="_generic__Skew__Matrix.html" title="make a generic skew symmetric matrix of variables">genericSkewMatrix</a> -- make a generic skew symmetric matrix of variables</span></span></li>
<li><span>genericSymmetricMatrix(Ring,RingElement,ZZ), see <span><a href="_generic__Symmetric__Matrix.html" title="make a generic symmetric matrix">genericSymmetricMatrix</a> -- make a generic symmetric matrix</span></span></li>
<li><span>genericSymmetricMatrix(Ring,ZZ), see <span><a href="_generic__Symmetric__Matrix.html" title="make a generic symmetric matrix">genericSymmetricMatrix</a> -- make a generic symmetric matrix</span></span></li>
<li><span><a href="_genus_lp__Ring_rp.html" title="arithmetic genus">genus(Ring)</a> -- arithmetic genus</span></li>
<li><span>GF(Ring), see <span><a href="___G__F.html" title="make a finite field">GF</a> -- make a finite field</span></span></li>
<li><span>heft(Ring), see <span><a href="_heft.html" title="heft vector of ring, module, graded module, or resolution">heft</a> -- heft vector of ring, module, graded module, or resolution</span></span></li>
<li><span>hilbertFunction(List,Ring), see <span><a href="_hilbert__Function.html" title="the Hilbert function">hilbertFunction</a> -- the Hilbert function</span></span></li>
<li><span>hilbertFunction(ZZ,Ring), see <span><a href="_hilbert__Function.html" title="the Hilbert function">hilbertFunction</a> -- the Hilbert function</span></span></li>
<li><span><a href="_hilbert__Polynomial_lp__Ring_rp.html" title="compute the Hilbert polynomial of the ring">hilbertPolynomial(Ring)</a> -- compute the Hilbert polynomial of the ring</span></li>
<li><span>Hom(Ideal,Ring), see <span><a href="___Hom_lp__Module_cm__Module_rp.html" title="module of homomorphisms">Hom(Module,Module)</a> -- module of homomorphisms</span></span></li>
<li><span>Hom(Module,Ring), see <span><a href="___Hom_lp__Module_cm__Module_rp.html" title="module of homomorphisms">Hom(Module,Module)</a> -- module of homomorphisms</span></span></li>
<li><span>Hom(Ring,Ideal), see <span><a href="___Hom_lp__Module_cm__Module_rp.html" title="module of homomorphisms">Hom(Module,Module)</a> -- module of homomorphisms</span></span></li>
<li><span>Hom(Ring,Module), see <span><a href="___Hom_lp__Module_cm__Module_rp.html" title="module of homomorphisms">Hom(Module,Module)</a> -- module of homomorphisms</span></span></li>
<li><span>icFracP(Ring), see <span><a href="../../IntegralClosure/html/_ic__Frac__P.html" title="compute the integral closure in prime characteristic">icFracP</a> -- compute the integral closure in prime characteristic</span></span></li>
<li><span>icFractions(Ring), see <span><a href="../../IntegralClosure/html/_ic__Fractions.html" title="fractions integral over an affine domain">icFractions</a> -- fractions integral over an affine domain</span></span></li>
<li><span>icMap(Ring), see <span><a href="../../IntegralClosure/html/_ic__Map.html" title="natural map from an affine domain into its integral closure">icMap</a> -- natural map from an affine domain into its integral closure</span></span></li>
<li><span><a href="_ideal_lp__Ring_rp.html" title="returns the defining ideal">ideal(Ring)</a> -- returns the defining ideal</span></li>
<li><span><a href="___Indexed__Variable_sp_us_sp__Ring.html" title="get a ring variable by name">IndexedVariable _ Ring</a> -- get a ring variable by name</span></li>
<li><span>isAffineRing(Ring), see <span><a href="_is__Affine__Ring.html" title="whether something is an affine ring">isAffineRing</a> -- whether something is an affine ring</span></span></li>
<li><span>isCommutative(Ring), see <span><a href="_is__Commutative.html" title="whether a ring is commutative">isCommutative</a> -- whether a ring is commutative</span></span></li>
<li><span>isField(Ring), see <span><a href="_is__Field.html" title="whether something is a field">isField</a> -- whether something is a field</span></span></li>
<li><span>isHomogeneous(Ring), see <span><a href="_is__Homogeneous.html" title="whether something is homogeneous (graded)">isHomogeneous</a> -- whether something is homogeneous (graded)</span></span></li>
<li><span>isNormal(Ring), see <span><a href="../../IntegralClosure/html/_is__Normal.html" title="determine if a reduced ring is normal">isNormal</a> -- determine if a reduced ring is normal</span></span></li>
<li><span>isQuotientOf(Ring,QuotientRing), see <span><a href="_is__Quotient__Of_lp__Ring_cm__Ring_rp.html" title="whether one ring is a quotient of another">isQuotientOf(Ring,Ring)</a> -- whether one ring is a quotient of another</span></span></li>
<li><span><a href="_is__Quotient__Of_lp__Ring_cm__Ring_rp.html" title="whether one ring is a quotient of another">isQuotientOf(Ring,Ring)</a> -- whether one ring is a quotient of another</span></li>
<li><span><a href="_is__Quotient__Of_lp__Type_cm__Ring_rp.html" title="whether one ring is a quotient of a ring of a given type">isQuotientOf(Type,Ring)</a> -- whether one ring is a quotient of a ring of a given type</span></li>
<li><span>isQuotientRing(Ring), see <span><a href="_is__Quotient__Ring.html" title="whether something is a quotient ring">isQuotientRing</a> -- whether something is a quotient ring</span></span></li>
<li><span>isRing(Ring), see <span><a href="_is__Ring.html" title="whether something is a ring">isRing</a> -- whether something is a ring</span></span></li>
<li><span>isSkewCommutative(Ring), see <span><a href="_is__Skew__Commutative.html" title="whether a ring has skew commuting variables">isSkewCommutative</a> -- whether a ring has skew commuting variables</span></span></li>
<li><span><a href="_jacobian_lp__Ring_rp.html" title="the Jacobian matrix of the polynomials defining a quotient ring">jacobian(Ring)</a> -- the Jacobian matrix of the polynomials defining a quotient ring</span></li>
<li><span>Constant ^ Ring, see <span><a href="_lift.html" title="lift to another ring">lift</a> -- lift to another ring</span></span></li>
<li><span>Number ^ Ring, see <span><a href="_lift.html" title="lift to another ring">lift</a> -- lift to another ring</span></span></li>
<li><span>RingElement ^ Ring, see <span><a href="_lift.html" title="lift to another ring">lift</a> -- lift to another ring</span></span></li>
<li><span>makeS2(Ring), see <span><a href="../../IntegralClosure/html/_make__S2.html" title="compute the S2ification of a reduced ring">makeS2</a> -- compute the S2ification of a reduced ring</span></span></li>
<li><span><a href="_map_lp__Ring_cm__Matrix_rp.html" title="make a ring map">map(Ring,Matrix)</a> -- make a ring map</span></li>
<li><span><a href="_map_lp__Ring_cm__Ring_rp.html" title="make a ring map, using the names of the variables">map(Ring,Ring)</a> -- make a ring map, using the names of the variables</span></li>
<li><span><a href="_map_lp__Ring_cm__Ring_cm__List_rp.html" title="make a ring map">map(Ring,Ring,List)</a> -- make a ring map</span></li>
<li><span><a href="_map_lp__Ring_cm__Ring_cm__Matrix_rp.html" title="make a ring map">map(Ring,Ring,Matrix)</a> -- make a ring map</span></li>
<li><span>map(Ring,Ring,RingMap), see <span><a href="_map_lp__Ring_cm__Ring_cm__Matrix_rp.html" title="make a ring map">map(Ring,Ring,Matrix)</a> -- make a ring map</span></span></li>
<li><span><a href="___Matrix_sp_st_st_sp__Ring.html" title="tensor product">Matrix ** Ring</a> -- tensor product</span></li>
<li><span>Ring ** Matrix, see <span><a href="___Matrix_sp_st_st_sp__Ring.html" title="tensor product">Matrix ** Ring</a> -- tensor product</span></span></li>
<li><span><a href="_matrix_lp__Ring_cm__List_rp.html" title="create a matrix from a doubly nested list of ring elements or matrices">matrix(Ring,List)</a> -- create a matrix from a doubly nested list of ring elements or matrices</span></li>
<li><span><a href="___Module_sp_st_st_sp__Ring.html" title="tensor product">Module ** Ring</a> -- tensor product</span></li>
<li><span>Ring ** Module, see <span><a href="___Module_sp_st_st_sp__Ring.html" title="tensor product">Module ** Ring</a> -- tensor product</span></span></li>
<li><span><a href="_module_lp__Ring_rp.html" title="">module(Ring)</a></span></li>
<li><span>multidegree(Ring), see <span><a href="_multidegree.html" title="multidegree">multidegree</a> -- multidegree</span></span></li>
<li><span><a href="_mutable__Identity_lp__Ring_cm__Z__Z_rp.html" title="make a mutable identity matrix">mutableIdentity(Ring,ZZ)</a> -- make a mutable identity matrix</span></li>
<li><span><a href="_mutable__Matrix_lp__Ring_cm__Z__Z_cm__Z__Z_rp.html" title="make a mutable matrix filled with zeroes">mutableMatrix(Ring,ZZ,ZZ)</a> -- make a mutable matrix filled with zeroes</span></li>
<li><span><a href="_numgens_lp__Ring_rp.html" title="number of generators of a polynomial ring">numgens(Ring)</a> -- number of generators of a polynomial ring</span></li>
<li><span><a href="_options_lp__Ring_rp.html" title="get values used for optional arguments">options(Ring)</a> -- get values used for optional arguments</span></li>
<li><span><a href="_poincare_lp__Ring_rp.html" title="assemble degrees of an ring into a polynomial">poincare(Ring)</a> -- assemble degrees of an ring into a polynomial</span></li>
<li><span>precision(Ring), see <span><a href="_precision.html" title="">precision</a></span></span></li>
<li><span><a href="___Proj_lp__Ring_rp.html" title="make a projective variety">Proj(Ring)</a> -- make a projective variety</span></li>
<li><span>Number _ Ring, see <span><a href="_promote.html" title="promote to another ring">promote</a> -- promote to another ring</span></span></li>
<li><span>RingElement _ Ring, see <span><a href="_promote.html" title="promote to another ring">promote</a> -- promote to another ring</span></span></li>
<li><span>random(List,Ring), see <span><a href="_random_lp__Type_rp.html" title="random element of a type">random(Type)</a> -- random element of a type</span></span></li>
<li><span>random(ZZ,Ring), see <span><a href="_random_lp__Type_rp.html" title="random element of a type">random(Type)</a> -- random element of a type</span></span></li>
<li><span><a href="___Ring_sp_sl_sp__Ideal.html" title="make a quotient ring">Ring / Ideal</a> -- make a quotient ring</span></li>
<li><span>Ring / List, see <span><a href="___Ring_sp_sl_sp__Ideal.html" title="make a quotient ring">Ring / Ideal</a> -- make a quotient ring</span></span></li>
<li><span>Ring / Module, see <span><a href="___Ring_sp_sl_sp__Ideal.html" title="make a quotient ring">Ring / Ideal</a> -- make a quotient ring</span></span></li>
<li><span>Ring / MonomialIdeal, see <span><a href="___Ring_sp_sl_sp__Ideal.html" title="make a quotient ring">Ring / Ideal</a> -- make a quotient ring</span></span></li>
<li><span>Ring / RingElement, see <span><a href="___Ring_sp_sl_sp__Ideal.html" title="make a quotient ring">Ring / Ideal</a> -- make a quotient ring</span></span></li>
<li><span>Ring / Sequence, see <span><a href="___Ring_sp_sl_sp__Ideal.html" title="make a quotient ring">Ring / Ideal</a> -- make a quotient ring</span></span></li>
<li><span>Ring / ZZ, see <span><a href="___Ring_sp_sl_sp__Ideal.html" title="make a quotient ring">Ring / Ideal</a> -- make a quotient ring</span></span></li>
<li><span><a href="___Ring_sp^_sp__List.html" title="make a free module">Ring ^ List</a> -- make a free module</span></li>
<li><span><a href="___Ring_sp^_sp__Z__Z.html" title="make a free module">Ring ^ ZZ</a> -- make a free module</span></li>
<li><span><a href="___Ring_sp_us_sp__List.html" title="make a monomial from a list of exponents">Ring _ List</a> -- make a monomial from a list of exponents</span></li>
<li><span><a href="___Ring_sp_us_sp__String.html" title="get a ring variable by name">Ring _ String</a> -- get a ring variable by name</span></li>
<li><span><a href="___Ring_sp_us_sp__Z__Z.html" title="get a ring variable by index">Ring _ ZZ</a> -- get a ring variable by index</span></li>
<li><span><tt>Ring _*</tt> (missing documentation<!-- tag: (_*,Ring) -->)</span></li>
<li><span><a href="___Ring_sp__Array.html" title="the standard way to make a polynomial ring">Ring Array</a> -- the standard way to make a polynomial ring</span></li>
<li><span><a href="___Ring_sp__List.html" title="make a local polynomial ring">Ring List</a> -- make a local polynomial ring</span></li>
<li><span><a href="___Ring_sp__Ordered__Monoid.html" title="make a polynomial ring">Ring OrderedMonoid</a> -- make a polynomial ring</span></li>
<li><span>Ring ~, see <span><a href="_sheaf_lp__Ring_rp.html" title="make a coherent sheaf of rings">sheaf(Ring)</a> -- make a coherent sheaf of rings</span></span></li>
<li><span><a href="_sheaf_lp__Ring_rp.html" title="make a coherent sheaf of rings">sheaf(Ring)</a> -- make a coherent sheaf of rings</span></li>
<li><span><a href="_sheaf_lp__Variety_cm__Ring_rp.html" title="make a coherent sheaf of rings">sheaf(Variety,Ring)</a> -- make a coherent sheaf of rings</span></li>
<li><span>singularLocus(Ring), see <span><a href="_singular__Locus.html" title="singular locus">singularLocus</a> -- singular locus</span></span></li>
<li><span><a href="___Spec_lp__Ring_rp.html" title="make an affine variety">Spec(Ring)</a> -- make an affine variety</span></li>
<li><span>substitute(Ideal,Ring), see <span><a href="_substitute.html" title="substituting values for variables">substitute</a> -- substituting values for variables</span></span></li>
<li><span>substitute(Matrix,Ring), see <span><a href="_substitute.html" title="substituting values for variables">substitute</a> -- substituting values for variables</span></span></li>
<li><span>substitute(Module,Ring), see <span><a href="_substitute.html" title="substituting values for variables">substitute</a> -- substituting values for variables</span></span></li>
<li><span>substitute(Number,Ring), see <span><a href="_substitute.html" title="substituting values for variables">substitute</a> -- substituting values for variables</span></span></li>
<li><span>substitute(RingElement,Ring), see <span><a href="_substitute.html" title="substituting values for variables">substitute</a> -- substituting values for variables</span></span></li>
<li><span>substitute(Vector,Ring), see <span><a href="_substitute.html" title="substituting values for variables">substitute</a> -- substituting values for variables</span></span></li>
<li><span><a href="___Symbol_sp_us_sp__Ring.html" title="get a ring variable by name">Symbol _ Ring</a> -- get a ring variable by name</span></li>
<li><span>symmetricAlgebra(Nothing,Ring,Matrix), see <span><a href="_symmetric__Algebra.html" title="the symmetric algebra of a module">symmetricAlgebra</a> -- the symmetric algebra of a module</span></span></li>
<li><span>symmetricAlgebra(Ring,Nothing,Matrix), see <span><a href="_symmetric__Algebra.html" title="the symmetric algebra of a module">symmetricAlgebra</a> -- the symmetric algebra of a module</span></span></li>
<li><span>symmetricAlgebra(Ring,Ring,Matrix), see <span><a href="_symmetric__Algebra.html" title="the symmetric algebra of a module">symmetricAlgebra</a> -- the symmetric algebra of a module</span></span></li>
<li><span><a href="_tensor_lp__Ring_cm__Ring__Map_cm__Matrix_rp.html" title="tensor product via a ring map">tensor(Ring,RingMap,Matrix)</a> -- tensor product via a ring map</span></li>
<li><span>tensor(Ring,RingMap,Module), see <span><a href="_tensor_lp__Ring_cm__Ring__Map_cm__Matrix_rp.html" title="tensor product via a ring map">tensor(Ring,RingMap,Matrix)</a> -- tensor product via a ring map</span></span></li>
<li><span>terms(Ring,RingElement), see <span><a href="_terms.html" title="provide a list of terms of a polynomial">terms</a> -- provide a list of terms of a polynomial</span></span></li>
<li><span><a href="_to__Field_lp__Ring_rp.html" title="declare that a ring is a field">toField(Ring)</a> -- declare that a ring is a field</span></li>
<li><span><a href="_use_lp__Ring_rp.html" title="install ring variables and ring operations">use(Ring)</a> -- install ring variables and ring operations</span></li>
<li><span><a href="_vars_lp__Ring_rp.html" title="row matrix of the variables">vars(Ring)</a> -- row matrix of the variables</span></li>
</ul>
<h2>Fixed objects of class Ring :</h2>
<ul><li><span><a href="___Q__Q.html" title="the class of all rational numbers">QQ</a> -- the class of all rational numbers</span></li>
<li><span><a href="___Z__Z.html" title="the class of all integers">ZZ</a> -- the class of all integers</span></li>
</ul>
</div>
<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="___Ring.html" title="the class of all rings">Ring</a> is <span>a <a href="___Type.html">type</a></span>, with ancestor classes <a href="___Type.html" title="the class of all types">Type</a> < <a href="___Mutable__Hash__Table.html" title="the class of all mutable hash tables">MutableHashTable</a> < <a href="___Hash__Table.html" title="the class of all hash tables">HashTable</a> < <a href="___Thing.html" title="the class of all things">Thing</a>.</p>
</div>
</div>
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