# Difference between revisions of "Unitary module"

From Encyclopedia of Mathematics

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− | A left (or right) [[ | + | {{TEX|done}}{{MSC|13C}} |

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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. [[ | + | 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