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Free Presentation Of A Module
In algebra, a free presentation of a module ''M'' over a commutative ring ''R'' is an exact sequence of ''R''-modules: :\bigoplus_ R \ \overset \to\ \bigoplus_ R \ \overset\to\ M \to 0. Note the image under ''g'' of the standard basis generates ''M''. In particular, if ''J'' is finite, then ''M'' is a finitely generated module. If ''I'' and ''J'' are finite sets, then the presentation is called a finite presentation; a module is called finitely presented if it admits a finite presentation. Since ''f'' is a module homomorphism between free modules, it can be visualized as an (infinite) matrix with entries in ''R'' and ''M'' as its cokernel. A free presentation always exists: any module is a quotient of a free module: F \ \overset\to\ M \to 0, but then the kernel of ''g'' is again a quotient of a free module: F' \ \overset \to\ \ker g \to 0. The combination of ''f'' and ''g'' is a free presentation of ''M''. Now, one can obviously keep "resolving" the kernels in this fashion; t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathematics), modules, vector spaces, lattice (order), lattices, and algebra over a field, algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variable (mathematics), variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form category (mathematics), mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
