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tensor products of rings

The operator ** or the function tensor can be used to construct tensor products of rings.
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
Other monomial orderings can be specified.
i2 : T = tensor(ZZ/101[x,y], ZZ/101[a,b], MonomialOrder => Eliminate 2)

o2 = T

o2 : PolynomialRing
The options to tensor can be discovered with options.
i3 : options tensor

o3 = OptionTable{Constants => false      }
                 DegreeLift => null
                 DegreeMap => null
                 DegreeRank => null
                 Degrees => null
                 Global => true
                 Heft => null
                 Inverses => null
                 Join => null
                 Local => false
                 MonomialOrder => null
                 MonomialSize => 32
                 SkewCommutative => {}
                 VariableBaseName => null
                 Variables => null
                 Weights => {}
                 WeylAlgebra => {}

o3 : OptionTable
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:
i4 : R = QQ[x,y]/(x^3-y^2);
i5 : T = R ** R

o5 = T

o5 : QuotientRing
i6 : generators T

o6 = {x, y, x, y}

o6 : List
i7 : {T_0 + T_1, T_0 + T_2}

o7 = {x + y, x + x}

o7 : List
We can change the variable names with the Variables option.
i8 : U = tensor(R,R,Variables => {x,y,x',y'})

o8 = U

o8 : QuotientRing
i9 : x + y + x' + y'

o9 = x + y + x' + y'

o9 : U