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Difference between revisions of "Completely-reducible module"

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A module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023990/c0239901.png" /> over an associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023990/c0239902.png" /> which can be represented as the sum of its irreducible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023990/c0239903.png" />-submodules (cf. [[Irreducible module|Irreducible module]]). Equivalent definitions are: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023990/c0239904.png" /> is the sum of its minimal submodules; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023990/c0239905.png" /> is isomorphic to a direct sum of irreducible modules; or 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023990/c0239906.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023990/c0239907.png" /> is a lattice with complements if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023990/c0239908.png" /> is completely reducible.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023990/c0239909.png" />-modules over a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023990/c02399010.png" /> are completely reducible, all left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023990/c02399011.png" />-modules are completely reducible as well, and vice versa; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023990/c02399012.png" /> is then said to be a completely-reducible ring or a [[Classical semi-simple ring|classical semi-simple ring]]. For a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023990/c02399013.png" /> to be completely reducible it is sufficient for it to be completely reducible when regarded as a left (right) module over itself.
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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|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=31723
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article