Let R <-* S be an embedding of associative noetherian rings such that 5 is finitely generated as a right Ä-module. There is a correspondence from the prime spectrum of S to the prime spectrum of R obtained by associating to a given prime ideal P of S the prime ideals of R minimal over P n R . The prime and primitive ideal theories for several specific noncommutative noetherian rings, including ...

We present a (random) polynomial-time algorithm to generate a randomGaussian integer with the uniform distribution among those with norm at most N , along with its prime factorization. The method generalizes to finding a random ideal in the ring of integers of a quadratic number field together with its prime ideal factorization. We also discuss the analogous problem for higher degree number fie...

In the p-th cyclotomic field Qpn , p a prime number, n ∈ N, the prime p is totally ramified and the only ideal above p is generated by ωn = ζpn − 1, with the primitive p-th root of unity ζpn = e 2πi pn . Moreover these numbers represent a norm coherent set, i.e. NQpn+1/Qpn(ωn+1) = ωn. It is the aim of this article to establish a similar result for the ray class field Kpn of conductor p over an ...

2. Primary ideals in C(X). C(X) denotes the ring of real-valued continuous functions on the topological space X, under ordinary pointwise addition and multiplication. By a primary ideal we mean an ideal which is contained in at most one maximal ideal. An O-ideal is an ideal / such that for each pair/i,/2G/ there exists eEI, depending on/i and f2, such that/<«=/.-, for i = l, 2 [7; 4]. A prime-l...

The ideal I 1 generated by the 2 2 quantum minors in the coordinate algebra of quantum matrices, O q (M m;n (k)), is investigated. Analogues of the First and Second Fundamental Theorems of Invariant Theory are proved. In particular, it is shown that I 1 is a completely prime ideal, that is, O q (M m;n (k))=I 1 is an integral domain, and that O q (M m;n (k))=I 1 is the ring of coinvariants of a ...

The ideal I1 generated by the 2× 2 quantum minors in the coordinate algebra of quantum matrices, Oq(Mm,n(k)), is investigated. Analogues of the First and Second Fundamental Theorems of Invariant Theory are proved. In particular, it is shown that I1 is a completely prime ideal, that is, Oq(Mm,n(k))/I1 is an integral domain, and that Oq(Mm,n(k))/I1 is the ring of coinvariants of a coaction of k[x...

Let R be a ring and let I 6= R be an ideal of R. Then I is said to be a completely prime ideal of R if R/I is a domain and is said to be completely semiprime if R/I is a reduced ring. In this paper, we introduce a new class of rings known as completely prime ideal rings. We say that a ring R is a completely prime ideal ring (CPI-ring) if every prime ideal of R is completely prime. We say that a...

In this paper, we define the notions of ultra and involution ideals in $BCK$-algebras. Then we get the relation among them and other ideals as (positive) implicative, associative, commutative and prime ideals. Specially, we show that in a bounded implicative $BCK$-algebra, any involution ideal is a positive implicative ideal and in a bounded positive implicative lower $BCK$-semilattice, the not...

In this work, we attempt to investigate the connection between various types of ideals (for examples (m,n)-ideal, bi-ideal, interior-ideal, quasi-ideal, prime-ideal and maximal-ideal) of an ordered semigroup (S, ·,≤) and the corresponding, hyperideals of its EL-hyperstructure (S, ∗) (if exists). Moreover, we construct the class of EL-Γ-semihypergroup, associated to a partially-ordered Γsemigroup.