tensor product of abelian groups


Group Theory

Monoidal categories



For AA and BB two abelian groups, their tensor product ABA \otimes B is a new abelian group which is such that a group homomorphism ABCA \otimes B \to C is equivalently a bilinear map out of AA and BB.



Let Ab be the collection of abelian groups, regarded as a multicategory whose multimorphisms are the multilinear maps A 1,,A nBA_1, \cdots, A_n \to B.

The tensor product A,BABA, B \mapsto A \otimes B in this multicategory is the tensor product of abelian groups.

Equivalently this means explicitly:


For A,BA, B two abelian groups, their tensor product of abelian groups is the abelian group ABA \otimes B which is the quotient of the free group on the product (direct sum) A×BA \times B by the relations

  • (a 1,b)+(a 2,b)(a 1+a 2,b)(a_1,b)+(a_2,b)\sim (a_1+a_2,b)

  • (a,b 1)+(a,b 2)(a,b 1+b 2)(a,b_1)+(a,b_2)\sim (a,b_1+b_2)

for all a,a 1,a 2Aa, a_1, a_2 \in A and b,b 1,b 2Bb, b_1, b_2 \in B.

In words: it is the group whose elements are presented by pairs of elements in AA and BB and such that the group operation for one argument fixed is that of the other group in the other argument.


The 0-ary relations (0,b)0(0,b)\sim 0 and (a,0)0(a,0)\sim 0 follow automatically; one needs them explicitly only if one generalises to abelian monoids.


By definition of the free construction and the quotient there is a canonical function of the underlying sets

A×BAB. A \times B \stackrel{\otimes}{\to} A \otimes B \,.

On elements this sends (a,b)(a,b) to the equivalence class that it represents under the above equivalence relations.

The following relates the tensor product to bilinear functions. It is a definition or a proposition dependening on whether one takes the notion of bilinear function to be defined before or after that of tensor product of abelian groups.


A function of underlying sets f:A×BCf : A \times B \to C is a bilinear function precisely if it factors by the morphism of 2 through a group homomorphism ϕ:ABC\phi : A \otimes B \to C out of the tensor product:

f:A×BABϕC. f : A \times B \stackrel{\otimes}{\to} A \otimes B \stackrel{\phi}{\to} C \,.


Monoidal category structure


Equipped with the tensor product \otimes of def. 2 Ab becomes a monoidal category.

The unit object in (Ab,)(Ab, \otimes) is the additive group of integers \mathbb{Z}.


To see that \mathbb{Z} is the unit object, consider for any abelian group AA the map

AA A \otimes \mathbb{Z} \to A

which sends for nn \in \mathbb{N} \subset \mathbb{Z}

(a,n)naa+a++a nsummands. (a, n) \mapsto n \cdot a \coloneqq \underbrace{a + a + \cdots + a}_{n\;summands} \,.

Due to the quotient relation defining the tensor product, the element on the left is also equal to

(a,n)=(a,1+1+1 nsummands)=(a,1)+(a,1)++(a,1) nsummands. (a, n) = (a, \underbrace{1 + 1 \cdots + 1}_{n\; summands}) = \underbrace{ (a,1) + (a,1) + \cdots + (a,1) }_{n\; summands} \,.

This shows that AAA \otimes \mathbb{Z} \to A is in fact an isomorphism.


The tensor product of abelian groups distributes over the direct sum of abelian groups

A sSB s sS(AB s). A \otimes \oplus_{s \in S} B_s \simeq \oplus_{s \in S} ( A \otimes B_s ) \,.



A monoid in (Ab,)(Ab, \otimes) is equivalently a ring.


Let (A,)(A, \cdot) be a monoid in (Ab,)(Ab, \otimes). The fact that the multiplication

:AAA \cdot : A \otimes A \to A

is bilinear means by the above that for all a 1,a 2,bAa_1, a_2, b \in A we have

(a 1+a 2)b=a 1b+a 2b (a_1 + a_2) \cdot b = a_1 \cdot b + a_2 \cdot b


b(a 1+a 2)=ba 1+ba 2. b \cdot (a_1 + a_2) = b \cdot a_1 + b \cdot a_2 \,.

This is precisely the distributivity law of the ring.


For nn \in \mathbb{N} positive we write n\mathbb{Z}_n for the cyclic group of order nn, as usual.


For a,ba,b \in \mathbb{N} and positive, we have

a b (a,b), \mathbb{Z}_a \otimes \mathbb{Z}_b \simeq \mathbb{Z}_{(a,b)} \,,

where (,)(-,-) denotes the greatest common divisor?.

A proof is spelled out for instance as (Conrad, theorem 4.1).


An exposition is in

  • Collin Roberts, Introduction to the tensor product (pdf)

and, in the further generality of the tensor product of modules, in

  • Keith Conrad, Tensor products (pdf)

Revised on August 21, 2015 02:59:59 by Urs Schreiber (