Editing Glossary of ring theory (section)

== M ==
{{glossary}}

{{term|1=minimal and maximal}}
{{defn|no=1|1=A left ideal ''M'' of the ring ''R'' is a [[maximal ideal|maximal left ideal]] (resp. minimal left ideal) if it is maximal (resp. minimal) among proper (resp. nonzero) left ideals. Maximal (resp. minimal) right ideals are defined similarly.}}
{{defn|no=2|1=A [[maximal subring]] is a subring that is maximal among proper subrings. A "minimal subring" can be defined analogously; it is unique and is called the [[characteristic subring]].}}

{{term|1=matrix}}
{{defn|no=1|1=A [[matrix ring]] over a ring ''R'' is a ring whose elements are square matrices of fixed size with the entries in ''R''. The matrix ring or the full matrix ring of matrices over ''R'' is ''the'' matrix ring consisting of all square matrices of fixed size with the entries in ''R''. When the grammatical construction is not workable, the term "matrix ring" often refers to the "full" matrix ring when the context makes no confusion likely; for example, when one says a semsimple ring is a product of matrix rings of division rings, it is implicitly assumed that "matrix rings" refer to "full matrix rings". Every ring is (isomorphic to) the full matrix ring over itself.}}
{{defn|no=2|1=The [[ring of generic matrices]] is the ring consisting of square matrices with entries in formal variables.}}

{{term|1=monoid}}
{{defn|1=A [[monoid ring]].}}

{{term|1=Morita}}
{{defn|1=Two rings are said to be [[Morita equivalent]] if the [[category of modules]] over the one is equivalent to the category of modules over the other.}}

{{glossary end}}