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- Quotient_module abstract "In abstract algebra, given a module and a submodule, one can construct their quotient module. This construction, described below, is analogous to how one obtains the ring of integers modulo an integer n, see modular arithmetic. It is the same construction used for quotient groups and quotient rings.Given a module A over a ring R, and a submodule B of A, the quotient space A/B is defined by the equivalence relation a ~ b if and only if b − a is in B,for any a and b in A. The elements of A/B are the equivalence classes [a] = { a + b : b in B }.The addition operation on A/B is defined for two equivalence classes as the equivalence class of the sum of two representatives from these classes; and in the same way for multiplication by elements of R. In this way A/B becomes itself a module over R, called the quotient module. In symbols, [a] + [b] = [a+b], and r·[a] = [r·a], for all a,b in A and r in R.".
- Quotient_module wikiPageID "364820".
- Quotient_module wikiPageRevisionID "541157707".
- Quotient_module hasPhotoCollection Quotient_module.
- Quotient_module subject Category:Module_theory.
- Quotient_module comment "In abstract algebra, given a module and a submodule, one can construct their quotient module. This construction, described below, is analogous to how one obtains the ring of integers modulo an integer n, see modular arithmetic. It is the same construction used for quotient groups and quotient rings.Given a module A over a ring R, and a submodule B of A, the quotient space A/B is defined by the equivalence relation a ~ b if and only if b − a is in B,for any a and b in A.".
- Quotient_module label "Module quotient".
- Quotient_module label "Moduł ilorazowy".
- Quotient_module label "Quotient module".
- Quotient_module label "Quotientenmodul".
- Quotient_module sameAs Quotientenmodul.
- Quotient_module sameAs Module_quotient.
- Quotient_module sameAs Moduł_ilorazowy.
- Quotient_module sameAs m.01_vyl.
- Quotient_module sameAs Q1432554.
- Quotient_module sameAs Q1432554.
- Quotient_module wasDerivedFrom Quotient_module?oldid=541157707.
- Quotient_module isPrimaryTopicOf Quotient_module.