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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From now on, if Y is a ring and x [member of] Y, we write <x> for the ideal of Y generated by x and write Y/<x> for the quotient ring of Y by <x>.
4 The RSA on the quotient ring of Gaussian integers
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).
Let R be a prime ring of characteristic different from 2, with extended centroid C, U its two-sided Utumi quotient ring, F [not equal to] 0 a nonzero generalized derivation of R, f([x.
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.
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.
Lee extended the definition of a generalized derivation as follows: an additive mapping F: J [right arrow] U such that F (xy) = F (x)y + xd(y), for all x, y [member of] J, where U is the right Utumi quotient ring of R, J is a dense right ideal of R and d is a derivation from J to 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),
Gordon, On the quotient ring by diagonal coinvariants, Invent.
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][.
One of three mathematicians among the top 10 winners, he entered a paper proposing new methods for studying polynomial quotient rings.

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