Difference between revisions of "Completely-reducible module"
From Encyclopedia of Mathematics
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| + | A module $M$ over an associative ring $R$ which can be represented as the sum of its irreducible $R$-submodules (cf. [[Irreducible module]]). Equivalent definitions are: 1) $M$ is the sum of its minimal submodules; 2) $M$ is isomorphic to a direct sum of irreducible modules; or 3) $M$ 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. | ||
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| + | 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==== | ====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> | ||
Latest revision as of 12:47, 13 December 2015
2020 Mathematics Subject Classification: Primary: 13C [MSN][ZBL]
semi-simple module
A module $M$ over an associative ring $R$ which can be represented as the sum of its irreducible $R$-submodules (cf. Irreducible module). Equivalent definitions are: 1) $M$ is the sum of its minimal submodules; 2) $M$ is isomorphic to a direct sum of irreducible modules; or 3) $M$ 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