Anneau sans diviseur de zéroEn théorie des anneaux, un anneau sans diviseur de zéro () est un anneau unitaire dans lequel un produit est nul seulement si l'un des facteurs est nul, autrement dit dans lequel l'implication suivante est vérifiée : En d'autres termes, c'est un anneau dans lequel il n'y a aucun diviseur de zéro (ni à droite, ni à gauche). Certains auteurs exigent également que la condition 1 ≠ 0 soit remplie ou, ce qui revient au même, que l'anneau ait au moins deux éléments.
Uniform moduleIn abstract algebra, a module is called a uniform module if the intersection of any two nonzero submodules is nonzero. This is equivalent to saying that every nonzero submodule of M is an essential submodule. A ring may be called a right (left) uniform ring if it is uniform as a right (left) module over itself. Alfred Goldie used the notion of uniform modules to construct a measure of dimension for modules, now known as the uniform dimension (or Goldie dimension) of a module.
Dimension theory (algebra)In mathematics, dimension theory is the study in terms of commutative algebra of the notion dimension of an algebraic variety (and by extension that of a scheme). The need of a theory for such an apparently simple notion results from the existence of many definitions of dimension that are equivalent only in the most regular cases (see Dimension of an algebraic variety).
Principal ideal ringIn mathematics, a principal right (left) ideal ring is a ring R in which every right (left) ideal is of the form xR (Rx) for some element x of R. (The right and left ideals of this form, generated by one element, are called principal ideals.) When this is satisfied for both left and right ideals, such as the case when R is a commutative ring, R can be called a principal ideal ring, or simply principal ring. If only the finitely generated right ideals of R are principal, then R is called a right Bézout ring.
Représentation régulièreEn mathématiques et plus précisément en théorie des groupes, les représentations régulières (gauche et droite) d'un groupe G sont les représentations de G associées aux deux actions (à gauche et à droite) de G sur lui-même par translation. Si G est un groupe fini ce sont, pour un corps fixé K, deux actions linéaires de G sur le K-espace vectoriel KG des applications de G dans K. Si G est un groupe localement compact, ce sont deux représentations continues unitaires de G sur un certain espace de Hilbert inclus dans CG.
Hopkins–Levitzki theoremIn abstract algebra, in particular ring theory, the Akizuki–Hopkins–Levitzki theorem connects the descending chain condition and ascending chain condition in modules over semiprimary rings. A ring R (with 1) is called semiprimary if R/J(R) is semisimple and J(R) is a nilpotent ideal, where J(R) denotes the Jacobson radical. The theorem states that if R is a semiprimary ring and M is an R-module, the three module conditions Noetherian, Artinian and "has a composition series" are equivalent.
Primitive ringIn the branch of abstract algebra known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector spaces and Weyl algebras over fields of characteristic zero. A ring R is said to be a left primitive ring if it has a faithful simple left R-module. A right primitive ring is defined similarly with right R-modules. There are rings which are primitive on one side but not on the other. The first example was constructed by George M.
Minimal idealIn the branch of abstract algebra known as ring theory, a minimal right ideal of a ring R is a non-zero right ideal which contains no other non-zero right ideal. Likewise, a minimal left ideal is a non-zero left ideal of R containing no other non-zero left ideals of R, and a minimal ideal of R is a non-zero ideal containing no other non-zero two-sided ideal of R . In other words, minimal right ideals are minimal elements of the partially ordered set (poset) of non-zero right ideals of R ordered by inclusion.
Hereditary ringIn mathematics, especially in the area of abstract algebra known as module theory, a ring R is called hereditary if all submodules of projective modules over R are again projective. If this is required only for finitely generated submodules, it is called semihereditary. For a noncommutative ring R, the terms left hereditary and left semihereditary and their right hand versions are used to distinguish the property on a single side of the ring.
Decomposition of a moduleIn abstract algebra, a decomposition of a module is a way to write a module as a direct sum of modules. A type of a decomposition is often used to define or characterize modules: for example, a semisimple module is a module that has a decomposition into simple modules. Given a ring, the types of decomposition of modules over the ring can also be used to define or characterize the ring: a ring is semisimple if and only if every module over it is a semisimple module.
