# quotient ring

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## 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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They cover the construction of a quotient ring of Z2F in which a binomial 1 + ?
Then [r.sub.p]([H.sub.n](1,q)) is isomorphic to the quotient ring Z[x,y]/1.
4.4 The RSA on the quotient ring of Gaussian integers
Hotta-Springer  and Garsia-Procesi  discovered that the cohomology ring of the Springer fiber indexed by a partition p of n is isomorphic to certain quotient ring of F[X], which admits a graded [G.sub.n]-module structure corresponding to the modified Hall-Littlewood symmetric function [[??].sub.[mu]] (x; t) via the Frobenius characteristic map.
Throught this paper we will use the following notation: U will be the (two-sided) Utumi quotient ring of a ring R (sometimes, as in , U is called the symmetric ring of quotients).
Throughout the paper unless specifically stated, R always denotes a prime ring with center Z(R) and extended centroid C, right Utumi quotient ring U.
Throughout this paper, unless specially stated, R always denotes a prime ring with center Z(R), with extended centroid C, and with two-sided Martin- dale quotient ring Q.
If S is a multiplicatively closed subset of R, then we may form the ring of quotients (or simply the quotient ring when there is no chance of confusion),
They are orders in Artinian rings, Goldie's theorem, and the largest left quotient ring of a ring; the invariant theory of Artin-Schelter regular algebras: a survey; the Catalan combinatorics of the hereditary Artin algebras; Grassmannians, flag varieties, and Gelfand-Zetlin polytopes; and on the combinatorics of the set of tilting modules.
Certain orbits of the reflection group action give a basis for the quotient ring, from this they found a formula for the corresponding Littlewood-Richardson coefficients in their quotient ring which was equal to a formula obtained by Kac  and Walton  for fusion coefficients in a Wess-Zumino-Witten conformal field theory.
Let R be a prime ring of characteristic [not equal to] 2 with right quotient ring U and extended centroid C, g [not equal to] 0 a generalized derivation of R, L a non-central Lie ideal of R and n [greater than or equal to] 1 such that [g(u), u][.sup.n] = 0, for all u [member of] L.
Gordon, On the quotient ring by diagonal coinvariants, Invent.

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