Editing Local ring (section)

=== General case===

The [[Jacobson radical]] ''m'' of a local ring ''R'' (which is equal to the unique maximal left ideal and also to the unique maximal right ideal) consists precisely of the non-units of the ring; furthermore, it is the unique maximal two-sided ideal of ''R''. However, in the non-commutative case, having a unique maximal two-sided ideal is not equivalent to being local.<ref>The 2 by 2 matrices over a field, for example, has unique maximal ideal {0}, but it has multiple maximal right and left ideals.</ref>

For an element ''x'' of the local ring ''R'', the following are equivalent:
* ''x'' has a left inverse
* ''x'' has a right inverse
* ''x'' is invertible
* ''x'' is not in ''m''.

If {{nowrap|(''R'', ''m'')}} is local, then the [[factor ring]] ''R''/''m'' is a [[skew field]]. If {{nowrap|''J'' ≠ ''R''}} is any two-sided ideal in ''R'', then the factor ring ''R''/''J'' is again local, with maximal ideal ''m''/''J''.

A [[Kaplansky's theorem on projective modules|deep theorem]] by [[Irving Kaplansky]] says that any [[projective module]] over a local ring is [[free module|free]], though the case where the module is finitely-generated is a simple corollary to [[Nakayama's lemma]]. This has an interesting consequence in terms of [[Morita equivalence]]. Namely, if ''P'' is a [[finitely generated module|finitely generated]] projective ''R'' module, then ''P'' is isomorphic to the free module ''R''<sup>''n''</sup>, and hence the ring of endomorphisms <math>\mathrm{End}_R(P)</math> is isomorphic to the full ring of matrices <math>\mathrm{M}_n(R)</math>. Since every ring Morita equivalent to the local ring ''R'' is of the form <math>\mathrm{End}_R(P)</math> for such a ''P'', the conclusion is that the only rings Morita equivalent to a local ring ''R'' are (isomorphic to) the matrix rings over ''R''.