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In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a module also generalizes the notion of an abelian group , since the abelian groups are exactly the modules over the ring of integers .
The projective dimension and the depth of a module over a commutative Noetherian local ring are complementary to each other. This is the content of the Auslander–Buchsbaum formula, which is not only of fundamental theoretical importance, but also provides an effective way to compute the depth of a module.
Given an ideal I in a commutative ring R and an R-module M, the direct sum = / + is a graded module over the associated graded ring / +. A morphism f : N → M {\displaystyle f:N\to M} of graded modules, called a graded morphism or graded homomorphism , is a homomorphism of the underlying modules that respects grading; i.e., f ( N i ) ⊆ M ...
2. The grade grade(M) of a module M over a ring R is grade(Ann M,R), which for a finitely generated module over a Noetherian ring is the smallest n such that Ext n R (M,R) is non-zero. 3. The grade of a module M over a Noetherian local ring with maximal ideal I is the grade of m on I. This is also called the depth of M. This is not consistent ...
One may thus replace the field of scalars by a ring R, and this gives the structure called a module over R, or R-module. The concepts of linear independence, span, basis, and linear maps (also called module homomorphisms ) are defined for modules exactly as for vector spaces, with the essential difference that, if R is not a field, there are ...
In 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.
Module theory is the branch of mathematics in which modules are studied. This is a glossary of some terms of the subject. ... ISBN 0-201-55540-9. Passman, ...
Time-keeping on this clock uses arithmetic modulo 12. Adding 4 hours to 9 o'clock gives 1 o'clock, since 13 is congruent to 1 modulo 12. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus.
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