Editing Module homomorphism (section)

== Examples ==
*The [[zero map]] ''M'' → ''N'' that maps every element to zero.
*A [[linear transformation]] between [[vector space]]s.
*<math>\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}/n, \mathbb{Z}/m) = \mathbb{Z}/\operatorname{gcd}(n,m)</math>.
*For a commutative ring ''R'' and [[Ideal (ring theory)|ideals]] ''I'', ''J'', there is the canonical identification
*:<math>\operatorname{Hom}_R(R/I, R/J) = \{ r \in R | r I \subset J \}/J</math>
:given by <math>f \mapsto f(1)</math>. In particular, <math>\operatorname{Hom}_R(R/I, R)</math> is the [[annihilator (ring theory)|annihilator]] of ''I''.
*Given a ring ''R'' and an element ''r'', let <math>l_r: R \to R</math> denote the left multiplication by ''r''. Then for any ''s'', ''t'' in ''R'',
*:<math>l_r(st) = rst = l_r(s)t</math>.
:That is, <math>l_r</math> is ''right'' ''R''-linear.
*For any ring ''R'',
**<math>\operatorname{End}_R(R) = R</math> as rings when ''R'' is viewed as a right module over itself. Explicitly, this isomorphism is given by the [[left regular representation]] <math>R \overset{\sim}\to \operatorname{End}_R(R), \, r \mapsto l_r</math>.
**Similarly, <math>\operatorname{End}_R(R) = R^{op}</math> as rings when ''R'' is viewed as a left module over itself. Textbooks or other references usually specify which convention is used.
**<math>\operatorname{Hom}_R(R, M) = M</math> through <math>f \mapsto f(1)</math> for any left module ''M''.<ref name=bourbaki/> (The module structure on Hom here comes from the right ''R''-action on ''R''; see [[#Module structures on Hom]] below.)
**<math>\operatorname{Hom}_R(M, R)</math> is called the [[dual module]] of ''M''; it is a left (resp. right) module if ''M'' is a right (resp. left) module over ''R'' with the module structure coming from the ''R''-action on ''R''. It is denoted by <math>M^*</math>.
*Given a ring homomorphism ''R'' → ''S'' of commutative rings and an ''S''-module ''M'', an ''R''-linear map θ: ''S'' → ''M'' is called a [[derivation (algebra)|derivation]] if for any ''f'', ''g'' in ''S'', {{nowrap|θ(''f g'') <nowiki>=</nowiki> ''f'' θ(''g'') + θ(''f'') ''g''}}.
*If ''S'', ''T'' are unital [[associative algebra]]s over a ring ''R'', then an [[algebra homomorphism]] from ''S'' to ''T'' is a [[ring homomorphism]] that is also an ''R''-module homomorphism.