Editing Injective hull (section)

==Properties==
* The injective hull of ''M'' is unique up to isomorphisms which are the identity on ''M'', however the isomorphism is not necessarily unique. This is because the injective hull's map extension property is not a full-fledged [[universal property]]. Because of this uniqueness, the hull can be denoted as ''E''(''M'').
* The injective hull ''E''(''M'') is a maximal [[essential extension]] of ''M'' in the sense that if ''M''⊆''E''(''M'') ⊊''B'' for a module ''B'', then ''M'' is not an essential submodule of ''B''.
* The injective hull ''E''(''M'') is a minimal injective module containing ''M'' in the sense that if ''M''⊆''B'' for an injective module ''B'', then ''E''(''M'') is (isomorphic to) a submodule of ''B''.
* If ''N'' is an essential submodule of ''M'', then ''E''(''N'')=''E''(''M'').
* Every module ''M'' has an injective hull. A construction of the injective hull in terms of homomorphisms Hom(''I'', ''M''), where ''I'' runs through the ideals of ''R'', is given by {{harvtxt|Fleischer|1968}}.
* The dual notion of a [[projective cover]] does ''not'' always exist for a module, however a [[flat cover]] exists for every module. 

===Ring structure===
In some cases, for ''R'' a subring of a self-injective ring ''S'', the injective hull of ''R'' will also have a ring structure.{{sfn|Lam|1999|loc=p. 78–80}} For instance, taking ''S'' to be a full [[matrix ring]] over a field, and taking ''R'' to be any ring containing every matrix which is zero in all but the last column, the injective hull of the right ''R''-module ''R'' is ''S''. For instance, one can take ''R'' to be the ring of all upper triangular matrices. However, it is not always the case that the injective hull of a ring has a ring structure, as an example in {{harv|Osofsky|1964}} shows.

A large class of rings which do have ring structures on their injective hulls are the [[nonsingular ring]]s.{{sfn|Lam|1999|loc=p. 366}} In particular, for an [[integral domain]], the injective hull of the ring (considered as a module over itself) is the [[field of fractions]]. The injective hulls of nonsingular rings provide an analogue of the ring of quotients for non-commutative rings, where the absence of the [[Ore condition]] may impede the formation of the [[classical ring of quotients]]. This type of "ring of quotients" (as these more general "fields of fractions" are called) was pioneered in {{harv|Utumi|1956}}, and the connection to injective hulls was recognized in {{harv|Lambek|1963}}.