Since the theory of ideals plays an important role in the theory of quotient semirings, in this paper, we will make an intensive study of the notions of Noetherian, Artinian, prime, primary, weakly primary and k-maximal ideals in commutative quotient semirings. The bulk of this paper is devoted to stating and proving analogues to several well-known theorems in the theory of the residue class ri...

lard and Schnorr algorithm. Farther factorization of prime quadratic integer π = ρρ in the ring 4 [ 2] ±  can be done similarly. Finite field r is represented as quotient ring 4 [ 2] /( ) ± ρ  . Integer exponent k is reduced in corresponding quotient ring by minimization of absolute value of its norm. Algorithms can be used for fast exponentiation in arbitrary cyclic group if its order can b...

Finite Frobenius rings have been characterized as precisely those finite rings satisfying the MacWilliams extension property, by work of Wood. In the present note we offer a generalization of this remarkable result to the realm of Artinian rings. Namely, we prove that a left Artinian ring has the left MacWilliams property if and only if it is left pseudo-injective and its finitary left socle em...

Let $R$ be a Noetherian local ring. We prove that is regular of dimension at most four if, and only every prime ideal, defining Gorenstein quotient ring, syzygetic. deduce characterization these rings in terms the Andr\'e-Quillen homology.

The goal of this paper is to give a definition generalization fuzzy prime $\Gamma $-ideals in $-rings by introducing 2-absorbing and weakly completely commutative their properties. Furthermore, we diagram which transition between definitions $-ring. Finally, introduce quotient $-ring $R$ induced the $-ideal $2$-absorbing

Abstract The classifying map of the integral Krichever–Hoehn formal group law is presented as a quotient Lazard ring by some explicit ideal.