Module indécomposableEn algèbre abstraite, un module est indécomposable s'il est non nul et ne peut pas être écrit comme une somme directe de deux sous-modules non nuls. L'indécomposabilité des modules est une notion plus faible que leur simplicité (qui est aussi parfois appelée irréductibilité). Une somme directe d'indécomposables est dite complètement décomposable, notion qui est donc plus faible que d'être semi-simple (somme directe de modules simples).
Gorenstein ringIn commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring R with finite injective dimension as an R-module. There are many equivalent conditions, some of them listed below, often saying that a Gorenstein ring is self-dual in some sense. Gorenstein rings were introduced by Grothendieck in his 1961 seminar (published in ). The name comes from a duality property of singular plane curves studied by (who was fond of claiming that he did not understand the definition of a Gorenstein ring).
Groupe de PrüferEn mathématiques, et plus particulièrement en théorie des groupes, on appelle p-groupe de Prüfer, ou encore groupe p-quasi-cyclique, pour un nombre premier p donné, tout groupe isomorphe au groupe multiplicatif formé par les racines complexes de l'unité dont les ordres sont des puissances de p. C'est donc un p-groupe abélien dénombrable. Les p-groupes de Prüfer étant isomorphes entre eux, on parle volontiers « du » p-groupe de Prüfer, sans en préciser un en particulier.
Perfect ringIn the area of abstract algebra known as ring theory, a left perfect ring is a type of ring in which all left modules have projective covers. The right case is defined by analogy, and the condition is not left-right symmetric; that is, there exist rings which are perfect on one side but not the other. Perfect rings were introduced in Bass's book. A semiperfect ring is a ring over which every finitely generated left module has a projective cover. This property is left-right symmetric.
BimoduleIn abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of mathematics, bimodules play a clarifying role, in the sense that many of the relationships between left and right modules become simpler when they are expressed in terms of bimodules. If R and S are two rings, then an R-S-bimodule is an abelian group such that: M is a left R-module and a right S-module.
Serial moduleIn abstract algebra, a uniserial module M is a module over a ring R, whose submodules are totally ordered by inclusion. This means simply that for any two submodules N1 and N2 of M, either or . A module is called a serial module if it is a direct sum of uniserial modules. A ring R is called a right uniserial ring if it is uniserial as a right module over itself, and likewise called a right serial ring if it is a right serial module over itself.
