Unitary module

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2020 Mathematics Subject Classification: Primary: 13C [MSN][ZBL]

A left (or right) module $M$ over a ring with 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
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Unitary module. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article