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Quotient ring - Wikipedia

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In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring[1] or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra.[2][3] It is a specific example of a quotient, as viewed from the general setting of universal algebra. Starting with a ring 𝑅 and a two-sided ideal 𝐼 in 𝑅 , a new ring, the quotient ring 𝑅 / 𝐼 , is constructed, whose elements are the cosets of 𝐼 in 𝑅 subject to special + and ⋅ operations. (Quotient ring notation always uses a fraction slash "/".) Quotient rings are distinct from the so-called "quotient field", or field of fractions, of an integral domain as well as from the more general "rings of quotients" obtained by localization. Given a ring 𝑅 and a two-sided ideal 𝐼 in 𝑅 , we may define an equivalence relation ∼ on 𝑅 as follows: Using the ideal properties, it is not difficult to check that ∼ is a congru

Quotient ring - Wikipedia Jump to content From Wikipedia, the free encyclopedia Reduction of a ring by one of its ideals Algebraic structure → Ring theory Ring theory Basic concepts Rings • Subrings • Ideal • Quotient ring • Fractional ideal • Total ring of fractions • Product of rings • Free product of associative algebras • Tensor product of algebras Ring homomorphisms • Kernel • Inner automorphism • Frobenius endomorphism Algebraic structures • Module • Associative algebra • Graded ring • Involutive ring • Category of rings • Initial ring Z {\displaystyle \mathbb {Z} } • Terminal ring 0 = Z

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