Difference between revisions of "Unitary module"
From Encyclopedia of Mathematics
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| − | A left (or right) [[ | + | |
| + | 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 $M$. | ||
====Comments==== | ====Comments==== | ||
| − | A unitary module as defined above is also (and better) called a unital module, [[#References|[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. [[ | + | A unitary module as defined above is also (and better) called a unital module, [[#References|[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., [[#References|[a3]]]. | Often the property that a module be unital is absorbed into the definition of a module, cf., e.g., [[#References|[a3]]]. | ||
Revision as of 22:17, 21 November 2014
2020 Mathematics Subject Classification: Primary: 13C [MSN][ZBL]
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 $M$.
Comments
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].
References
| [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 |
How to Cite This Entry:
Unitary module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unitary_module&oldid=34755
Unitary module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unitary_module&oldid=34755
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article