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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_rings.html" title="">rings</a> > <a href="_tensor_spproducts_spof_springs.html" title="">tensor products of rings</a></div>
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<div><h1>tensor products of rings</h1>
<div>The operator <a href="__st_st.html" title="a binary operator, usually used for tensor product or Cartesian product">**</a> or the function <a href="_tensor.html" title="tensor product">tensor</a> can be used to construct tensor products of rings.<table class="examples"><tr><td><pre>i1 : ZZ/101[x,y]/(x^2-y^2) ** ZZ/101[a,b]/(a^3+b^3)
ZZ
---[x, y, a, b]
101
o1 = ------------------
2 2 3 3
(x - y , a + b )
o1 : QuotientRing</pre>
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Other monomial orderings can be specified.<table class="examples"><tr><td><pre>i2 : T = tensor(ZZ/101[x,y], ZZ/101[a,b], MonomialOrder => Eliminate 2)
o2 = T
o2 : PolynomialRing</pre>
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The options to <tt>tensor</tt> can be discovered with <a href="_options.html" title="get options">options</a>.<table class="examples"><tr><td><pre>i3 : options tensor
o3 = OptionTable{DegreeLift => null }
DegreeMap => null
DegreeRank => null
Degrees => null
Global => true
Heft => null
Inverses => false
Join => null
Local => false
MonomialOrder => null
MonomialSize => 32
SkewCommutative => {}
VariableBaseName => null
Variables => null
Weights => {}
WeylAlgebra => {}
o3 : OptionTable</pre>
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Given two (quotients of) polynomial rings, say, R = A[x1, ..., xn]/I, S = A[y1,...,yn]/J, then R ** S = A[x1,...,xn,y1, ..., yn]/(I + J). The variables in the two rings are always considered as different. If they have name conflicts, you may still use the variables with indexing, but the display will be confusing:<table class="examples"><tr><td><pre>i4 : R = QQ[x,y]/(x^3-y^2);</pre>
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<tr><td><pre>i5 : T = R ** R
o5 = T
o5 : QuotientRing</pre>
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<tr><td><pre>i6 : generators T
o6 = {x, y, x, y}
o6 : List</pre>
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<tr><td><pre>i7 : {T_0 + T_1, T_0 + T_2}
o7 = {x + y, x + x}
o7 : List</pre>
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We can change the variable names with the <a href="___Variables.html" title="name for an optional argument">Variables</a> option.<table class="examples"><tr><td><pre>i8 : U = tensor(R,R,Variables => {x,y,x',y'})
o8 = U
o8 : QuotientRing</pre>
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<tr><td><pre>i9 : x + y + x' + y'
o9 = x + y + x' + y'
o9 : U</pre>
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