View source for Completely-reducible module

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.

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.

====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>