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Difference between revisions of "Unitary module"

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A left (or right) [[Module|module]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095550/u0955501.png" /> over a ring with an identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095550/u0955502.png" /> such that multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095550/u0955503.png" /> is the identity operator, i.e. the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095550/u0955504.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095550/u0955505.png" /> for right modules), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095550/u0955506.png" />, is the identity automorphism of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095550/u0955507.png" />.
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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 $M$.
  
  
  
 
====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. [[Unitary space|Unitary space]].
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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]]].

Latest revision as of 16:19, 11 September 2016

2010 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 $M$.


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=14610
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article