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# tensor(Module,Module) -- tensor product of modules

## Synopsis

• Function: tensor
• Usage:
M ** N
tensor(M, N)
• Inputs:
• M, ,
• N, ,
• Outputs:
• , the tensor product of $M$ and $N$

## Description

If $M$ has generators $m_1,$m_2, \dots, $m_r$, and $N$ has generators $n_1, n_2, \dots, n_s$, then $M \otimes N$ has generators $m_i\otimes n_j$ for $0<i\leq r$ and $0<j\leq s$.

 i1 : R = ZZ[a..d]; i2 : M = image matrix {{a,b}} o2 = image | a b | 1 o2 : R-module, submodule of R i3 : N = image matrix {{c,d}} o3 = image | c d | 1 o3 : R-module, submodule of R i4 : M ** N o4 = cokernel {2} | -d 0 -b 0 | {2} | c 0 0 -b | {2} | 0 -d a 0 | {2} | 0 c 0 a | 4 o4 : R-module, quotient of R i5 : N ** M o5 = cokernel {2} | -b 0 -d 0 | {2} | a 0 0 -d | {2} | 0 -b c 0 | {2} | 0 a 0 c | 4 o5 : R-module, quotient of R

Use trim or minimalPresentation if a more compact presentation is desired.

Use flip to produce the isomorphism $M \otimes N \to N \otimes M$.

To recover the factors from the tensor product, use the function formation.