The total quotient ring of the ring of holomorphic functions on an open set " D " of complex numbers is the ring of meromorphic functions on " D ", even if " D " is not connected.
32.
The notation \ mathbb { Z } / n \ mathbb { Z } is used, because it is the quotient ring of \ mathbb { Z } by the field when n \ mathbb { Z } is a maximal ideal, that is, when is prime.
33.
A simple algebraic extension of a field, generated by the root of an irreducible polynomial of degree may be identified to the quotient ring K [ X ] / \ langle p \ rangle,, and its elements are in Euclidean division by of the product of polynomials.
34.
The notion of localization of a ring ( in particular the localization with respect to a prime ideal, the localization consisting in inverting a single element and the total quotient ring ) is one of the main differences between commutative algebra and the theory of non-commutative rings.
35.
A RLWE-SIG works in the quotient ring of polynomials modulo a degree n polynomial ? ( x ) with coefficients in the finite field Z q for an odd prime q ( i . e . the ring Z q [ x ] / ? ( x ) ).
36.
Another way to view this last example is to note that i is a ideal ( " X " 2 + 1 ) is generated by a polynomial irreducible over "'R "', the ideal is maximal, hence the quotient ring is a field.
37.
The intimate relationship between ring homomorphisms, kernels and quotient rings can be summarized as follows : " the ring homomorphisms defined on R / I are essentially the same as the ring homomorphisms defined on R that vanish ( i . e . are zero ) on I ".
38.
Principal right ideal rings and right B�zout rings are also closed under quotients, that is, if " I " is a proper ideal of principal right ideal ring " R ", then the quotient ring " R / I " is also principal right ideal ring.
39.
For nonassociative rings, the definition of a two-sided ideal needed to define a quotient ring is identical to that for a noncommutative ring ( although the ideal generated by an element or subset of a nonassociative ring is a much worse monster ) .-- talk ) 17 : 01, 8 April 2012 ( UTC)
40.
In yet a simpler way, we may think of the Jacobson radical of a ring as method to " mod out bad elements " of the ring that is, members of the Jacobson radical act as 0 in the quotient ring, " R " / " J " ( " R " ).
