Semiprime Ideals Research Articles

On semiprime ideals in lattices

The basic aim of this note is to show that alleles can be a useful tool for investigations of semiprime ideals. By means of alleles we characterize semiprime ideals in general lattices. We also study when a sectionally complemented lattice is distributive.

Semiprime ideals in general lattices

The ring-theoretical concept of semiprime ideal is appropriately defined for lattices. We prove that an ideal I of a lattice L is semiprime iff I is the kernel of some homomorphisms of L onto a distributive lattice with zero. As a corollary we generalize the Prime Separation Theorem to arbitrary lattices. The theory of semiprime ideals is developed here without assuming the axiom of choice. The radical of an ideal is defined as its semiprime closure and we show that the Ultrafilter Principle is equivalent to the statement that every semiprime ideal is representable as an intersection of prime ideals.

COMPLETELY PRIME AND COMPLETELY SEMIPRIME IDEALS IN Γ-RLNGS

Abstract In this note we introduce the concept of completely prime and completely semiprime ideals for Γ-rings and study some of the properties of these ideals. Furthermore we introduce the concept of the completely prime radical of an ideal of a Γ-ring and also show that the upper radical determined by a class of completely prime rings is a special radical. We define an mc-system to give an elementwise characterization of the completely prime radical. Finally we show that if R is the right operator ring of a Γ ring H and W has a strong right unity, then there is a one-to-one correspondence between the completely prime ideals of R and that of M.

Completely semiprime and Abelian regular ideals of a ring

Completely semiprime and Abelian regular ideals of a ring

Techniques Maketh O.R.

The title of this paper is meant to draw the reader's attention to the commonly held view that is a collection of techniques. Although many members ofthe Society would wish to take issue with this concept, they themselves must take some responsibility that this impression is still held. There has been no coherent development of the subject, and the definition that O.R. is what workers do has enabled anything to be classed as If an worker was fascinated with the problem in the generation of random numbers, then that was, and is O.R.; likewise so are polyhedral polarity, polytopes, stochastic dominance, fuzzy semi-prime ideals, S-cyclic Steiner systems etc. No clear-cut agreement concerning a definition of simply perpetuates the growth of very specialized interests. Outsiders who are not party to the reasons for the narrow technical developments have latched on to the more comprehensible parts of our activities and frequently assumed that the deflnable technical areas represent the whole of the O.R.'s contribution, influence and value. One cause (blame?) must be the tendency for academics to dominate the subject. In 1960 there were virtually no members ofthe U.K. Society in universities, but in 1983 about one quarter gave their addresses as an educational institution. Similarly, in America the proportion has risen from about 10 to 45%. The other significant feature of this growth is that the organisers of the societies have been drawn more and more from these academics. It is not necessarily a bad thing that it is the 'research' side of a body which is also the group that has a major say in its operations. Problems are likely only if the group tries to impose its interests on the majority of the members. Many professional institutions rely on their teaching members to provide the nucleus of their committee and administrative structure. Certainly, the engineering, medical and legal institutions use these members to the full. The view that is different and should not be dominated by academic needs has been maintained in the U.K. and by a few groups elsewhere. It is unfortunate that any suggestion of planned entry to or performance standards being set is greeted with howls of elitism or cries of subversion. Dominance by one group is important only if that dominance directs the course of the Society against the best interests of its members. Experience in O.R.?at least in the U.K. and through IFORS?suggests that great efforts are being made to present the subject in terms of its use, not its theory. Changes in emphasis of some publications of recent Presidents of IFORS, show how some academics are biassed away from theory and are advocating the application image of The techniques view of the subject has grown up for a variety of reasons, and this paper will attempt to discuss some of them and to consider some implications.

Some remarks on topologically semiprime ideals

Some remarks on topologically semiprime ideals

A characterization of semi-prime ideals in near-rings

AbstractIt is well-known that in any near-ring, any intersection of prime ideals is a semi-prime ideal. The aim of this note is to prove that any ideal is a prime ideal if and only if it is equal to its prime radical. As a consequence of this we have any semi-prime ideal I in a near-ring N is the intersection of minimal prime ideals of I in N and that I is the intersection of all prime ideals containing I.

Idempotents and completely semiprime ideals

Idempotents and completely semiprime ideals

Fuzzy semiprime ideals in semigroups

Abstract In this paper we shall introduce the notion of fuzzy semiprimality in a semigroup, which is an extension of semiprimality in it, and characterize a semigroup that is a semilattice of simple semigroups in terms of fuzzy semiprimality.

A Characterization of semiprime ideals in near-rings

AbstractIt is well known that in any near-ring, any intersection of prime ideals is a semiprime ideal. The aim of this paper is to prove that any semiprime ideal I in a near-ring N is the intersection of all minimal prime ideals of I in N. As a consequence of this we have any seimprime ideal I is the intersectionof all prime ideals containing I.

Radicals and semi-prime ideals of jordan triple systems

Radicals and semi-prime ideals of jordan triple systems

Factoring Ideals into Semiprime Ideals

Let D be an integral domain with 1 ≠ 0 . We consider “property SP” in D, which is that every ideal is a product of semiprime ideals. (A semiprime ideal is equal to its radical.) It is natural to consider property SP after studying Dedekind domains, which involve factoring ideals into prime ideals. We prove that a domain D with property SP is almost Dedekind, and we give an example of a nonnoetherian almost Dedekind domain with property SP.

Semiprime ideals in rings with finite group actions

Let R be a ring, G a finite group of automorphisms acting on R, and R G the-fixed subring of R. We prove that if R is semiprime with no additive ¦ G¦-torsion, then R is left Goldie if and only if R G is left Goldie. By coupling this with an examination of the prime ideal structures of R G and R, we are able to prove that if ¦G ¦ is invertible in R and R G is left Noetherian, then R satisfies the-ascending chain condition on semiprime ideals, every semiprime factor ring of R is left Goldie, and nil subrings of R are nilpotent. For the pair R G and R, we also consider various other properties of prime and maximal ideals such as lying over, going up, going down, and incomparability.

T-torsion theories and central localizations

We use the notion of a critical right ideal as developed by Lambek and Michler [9] as a generalization of prime ideals in a commutative ring. Call a right ideal P of the ring R critical if for every H properly containing P, Hom(R/H, E(r/P)) = 0 where E(R/P) is the injective hull of R/P (Gordon and Robson call R/P monoform). In the first section we look at T-torsion theories (T in the sense of Goldman [4]). Let S be a ring of quotients with respect to some T-torsion theory. Then there is a natural bijection between critical right ideals of S and closed critical right ideals of R (P closed means R/P torsion-free). We also prove that this torsion theory is cogenerated by a family of critical right ideals. A corollary of this gives a generalization of a result of Richman [12]. Namely, we show that if R is prime Goldie and S is a left flat, epimorphic ring extension of R, then S = n RP where P runs over some family of critical right ideals. Here R, denotes the ring of quotients with respect to the torsion theor! cogenerated by E(R/P). In the second section we look at localizations of R at prime ideals of the Center R = -4. In the most general case, let R be an arbitrary ring (with identity). Then w. dim R = sup w. dim R./l , where ;K runs through the maximal ideals of .-I. (w. dim R is weak or flat dimension of R.) \Ve then investigate the case when R either is (i) finitely generated over A (as a module) or (ii) R is Xoetherian and is integral over A. If R satisfies either of these conditions, then there is a canonical bijection between (two-sided) prime ideals of R and the equivalence classes of critical right ideals (PI is equivalent to P2 if E(R/P,) N E(R/P,)). Theorem 2.14 and Corollary 2.15 give the main results of this paper, which are as follows: If R satisfies either (i) or (ii), then there exists a family F of semiprime ideals such that: (I) for each N E 7, localization at N is T; NR, = Jacobson radical of R, ; and R,/NR, is Artinian; and (2) an R module M is flat iff MA, is R,\flat for all NE .F.

Radicals and semiprime ideals in semigroups

Radicals and semiprime ideals in semigroups

Localization in Non-Commutative Noetherian Rings

To construct a well behaved localization of a noetherian ringRat a semiprime ideal S, it seems necessary to assume that the set(S)of moduloSregular elements satisfies the Ore condition ; and it is convenient to require the Artin Rees property for the Jacobson radical of the quotient ringRsin addition: one calls such 5classical.To determine the classical semiprime ideals is no easy matter; it happens frequently that a prime ideal fails to be classical itself, but is minimal over a suitable classical semiprime ideal.

Another Characterization of Semiprime Ideals in C(X)

Another Characterization of Semiprime Ideals in C(X)

Localization at semiprime ideals

Localization at semiprime ideals

Localization of Right Noetherian Rings at Semiprime Ideals

In [11] and [12] we investigated the process of localization of right Noetherian rings R at prime ideals. We shall now extend these investigations to semiprime ideals N of R.In Section 2 we show that localizing at the injective right R-module E(R/N) is the same as localizing with respect to the multiplicative set

Extension of results concerning rings in which semi-primary ideals are primary

Extension of results concerning rings in which semi-primary ideals are primary