Let be a ring and left modules over . A function is said to be a left module homomorphism![Mathworld](http://mathworld.wolfram.com/favicon_mathworld.png)
![Planetmath](http://planetmath.org/sites/default/files/fab-favicon.ico)
![Planetmath](http://planetmath.org/sites/default/files/fab-favicon.ico)
![Planetmath](http://planetmath.org/sites/default/files/fab-favicon.ico)
![Planetmath](http://planetmath.org/sites/default/files/fab-favicon.ico)
![Planetmath](http://planetmath.org/sites/default/files/fab-favicon.ico)
![Planetmath](http://planetmath.org/sites/default/files/fab-favicon.ico)
(over ) if
-
1.
is additive: , and
-
2.
If are right -modules, then is a right module homomorphism provided that is additive and preserves right scalar multiplication: for any . If is commutative![Planetmath](http://planetmath.org/sites/default/files/fab-favicon.ico)
![Planetmath](http://planetmath.org/sites/default/files/fab-favicon.ico)
![Planetmath](http://planetmath.org/sites/default/files/fab-favicon.ico)
, any left module homomorphism is a right module homomorphism, and vice versa, and we simply call a module homomorphism![Mathworld](http://mathworld.wolfram.com/favicon_mathworld.png)
ZZR,SM,N(R,S)f:M→NfRRMRNSSMSN(Z,Z)R(R,Z)S(Z,S)