Difference between revisions of "Unitary module"
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A left (or right) [[Module|module]]
A left (or right) [[Module|module]] over a ring with an identity such that multiplication by is the identity operator, i.e. the transformation (respectively, for right modules), , is the identity automorphism of the group .
Revision as of 17:31, 11 April 2014
A left (or right) module $M$ over a ring with an identity $e$ such that multiplication by $e$ is the identity operator, i.e. the transformation $m\to em$ (respectively, $m\to me$ for right modules), $m\in M$, is the identity automorphism of the group $M$.
A unitary module as defined above is also (and better) called a unital module, [a1]. The terminology "unitary module" can cause confusion in that it may suggest some module generalization of the concept of a unitary vector space, cf. Unitary space.
Often the property that a module be unital is absorbed into the definition of a module, cf., e.g., [a3].
|[a1]||P.M. Cohn, "Algebra" , 1–2 , Wiley (1991) pp. 409|
|[a2]||O. Zariski, P. Samuel, "Commutative algebra" , 1 , v. Nostrand (1958) pp. 134|
|[a3]||H. Matsumura, "Commutative ring theory" , Cambridge Univ. Press (1989) pp. 7|
Unitary module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unitary_module&oldid=14610