Difference between revisions of "Completely-reducible module"
From Encyclopedia of Mathematics
(Importing text file) |
(TeX) |
||
| Line 1: | Line 1: | ||
| − | A module | + | {{TEX|done}} |
| + | A module $A$ over an associative ring $R$ which can be represented as the sum of its irreducible $R$-submodules (cf. [[Irreducible module|Irreducible module]]). Equivalent definitions are: 1) $A$ is the sum of its minimal submodules; 2) $A$ is isomorphic to a direct sum of irreducible modules; or 3) $A$ coincides with its [[Socle|socle]]. A submodule and a quotient module of a completely-reducible module are also completely reducible. The lattice of submodules of a module $M$ is a lattice with complements if and only if $M$ is completely reducible. | ||
| − | If all right | + | If all right $R$-modules over a ring $R$ are completely reducible, all left $R$-modules are completely reducible as well, and vice versa; $R$ is then said to be a completely-reducible ring or a [[Classical semi-simple ring|classical semi-simple ring]]. For a ring $R$ to be completely reducible it is sufficient for it to be completely reducible when regarded as a left (right) module over itself. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J. Lambek, "Lectures on rings and modules" , Blaisdell (1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> J. Lambek, "Lectures on rings and modules" , Blaisdell (1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)</TD></TR></table> | ||
Revision as of 14:43, 15 April 2014
A module $A$ over an associative ring $R$ which can be represented as the sum of its irreducible $R$-submodules (cf. Irreducible module). Equivalent definitions are: 1) $A$ is the sum of its minimal submodules; 2) $A$ is isomorphic to a direct sum of irreducible modules; or 3) $A$ coincides with its socle. A submodule and a quotient module of a completely-reducible module are also completely reducible. The lattice of submodules of a module $M$ is a lattice with complements if and only if $M$ is completely reducible.
If all right $R$-modules over a ring $R$ are completely reducible, all left $R$-modules are completely reducible as well, and vice versa; $R$ is then said to be a completely-reducible ring or a classical semi-simple ring. For a ring $R$ to be completely reducible it is sufficient for it to be completely reducible when regarded as a left (right) module over itself.
References
| [1] | J. Lambek, "Lectures on rings and modules" , Blaisdell (1966) |
| [2] | N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956) |
How to Cite This Entry:
Completely-reducible module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Completely-reducible_module&oldid=15557
Completely-reducible module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Completely-reducible_module&oldid=15557
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article