# quotient ring

(redirected from Quotient algebra)

## quotient ring

[′kwō·shənt ‚riŋ]
(mathematics)
A ring R / I whose elements are the cosets rI of a given ideal I in a given ring R, where the additive and multiplicative operations have the form: r1 I + r2 I ≡ (r1+ r2) I and r1 I · r2 I ≡ (r1· r2) I. Also known as factor ring; residue class ring.
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Note that a two-sided ideal I is regular if the quotient algebra A/I has [a] = a + I as a unit element.
This leads the description of the projective class ring [r.sub.p]([H.sub.n](q)), the Jacobson radical J([R.sub.p]([H.sub.n](q))) of the projective class algebra [R.sub.p] ([H.sub.n](q)) and the quotient algebra [R.sub.p] ([H.sub.n] (q))/ J ([R.sub.p] ([H.sub.n] (q))).
Then there exists a homogeneous [H.sub.n](0)-invariant ideal [I.sub.[alpha]] of the multigraded algebra F[[B.sub.n]] such that the quotient algebra F[[B.sub.n]]/[I.sub.[alpha]] becomes a projective [H.sub.n](0)-module with multigraded noncommutative characteristic equal to
Let L([H.sub.1](X)) be the Lie algebra over Z defined to be the quotient algebra of T([H.sub.1](X)) by the ideal I generated by elements of the form
Filters are closely related with lattice valued logics, they are also particularly interesting because they closely related to congruence relation, with any filter we can associate a quotient algebra. If we want such a quotient algebra to satisfy particular properties, we need to put extra conditions on the filter.
In this case, the quotient algebra A/k(A) is a nonzero unital Abelian [C.sup.*]-algebra.
equivalently, those ideals J [subset] k[x] for which G spans the quotient algebra k[x]/J.
A topological algebra A over C is a Gelfand-Mazur algebra if the quotient algebra A/M is topologically isomorphic to C for every M [member of] m(A).
Let [??] = Fin(M)/I be the quotient algebra. We let [??] = x + I.
However, there are several complications to be overcome: (i) the definition of the join matrix, (ii) the choice of the finiteness condition on the join matrices, and (iii) the choice of the definition of the quotient algebra.
Then all non-zero elements in the quotient algebra A/M are topologically invertible.
For example, the algebra QSym is the quotient algebra of FQSym where [F.sub.[sigma]] and [F.sub.[tau]] are identified if [sigma] and [tau] have the same descents.

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