Noetherian moduleIn abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion. Historically, Hilbert was the first mathematician to work with the properties of finitely generated submodules. He proved an important theorem known as Hilbert's basis theorem which says that any ideal in the multivariate polynomial ring of an arbitrary field is finitely generated.
Radical of a moduleIn mathematics, in the theory of modules, the radical of a module is a component in the theory of structure and classification. It is a generalization of the Jacobson radical for rings. In many ways, it is the dual notion to that of the socle soc(M) of M. Let R be a ring and M a left R-module. A submodule N of M is called maximal or cosimple if the quotient M/N is a simple module. The radical of the module M is the intersection of all maximal submodules of M, Equivalently, These definitions have direct dual analogues for soc(M).
Algèbre de type finiEn algèbre commutative, la notion d'algèbre de type fini est une première généralisation des anneaux de polynômes à un nombre fini d'indéterminées. Ces algèbres possèdent de bonnes propriétés relatives à l'anneau de base, et de bonnes propriétés absolues lorsque l'anneau de base est un corps. Les algèbres de type fini sur un corps sont les objets algébriques de base des variétés algébriques. Sur un corps k, attention à ne pas confondre une algèbre de type fini avec une extension de type fini qui n'est jamais de type fini en tant que k-algèbre sauf si c'est une extension finie.
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.
Projective coverIn the branch of abstract mathematics called , a projective cover of an object X is in a sense the best approximation of X by a projective object P. Projective covers are the of injective envelopes. Let be a and X an object in . A projective cover is a pair (P,p), with P a projective object in and p a superfluous epimorphism in Hom(P, X). If R is a ring, then in the category of R-modules, a superfluous epimorphism is then an epimorphism such that the kernel of p is a superfluous submodule of P.
Smith normal formIn mathematics, the Smith normal form (sometimes abbreviated SNF) is a normal form that can be defined for any matrix (not necessarily square) with entries in a principal ideal domain (PID). The Smith normal form of a matrix is diagonal, and can be obtained from the original matrix by multiplying on the left and right by invertible square matrices. In particular, the integers are a PID, so one can always calculate the Smith normal form of an integer matrix.
Torsionless moduleIn abstract algebra, a module M over a ring R is called torsionless if it can be embedded into some direct product RI. Equivalently, M is torsionless if each non-zero element of M has non-zero image under some R-linear functional f: This notion was introduced by Hyman Bass. A module is torsionless if and only if the canonical map into its double dual, is injective. If this map is bijective then the module is called reflexive. For this reason, torsionless modules are also known as semi-reflexive.
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.
