Multiplicative Ideal Theory Research Articles

A Note on Multiplicative Ideal Theory

We state a semigroup version of (19.15)THEOREM and (36.2)THEOREM, and disprove (30.9)THEOREM of Gilmer’s book [3] for commutative semigroups. We give a condition for the Kronecker function ring D* with respect to a star-operation * on an integral domain D to be well-defined.

A Note on Multiplicative Ideal Theory

A Note on Multiplicative Ideal Theory

Dilworth's principal elements

Principal elements were introduced in multiplicative lattices by R. P. Dilworth, following an earlier but less successful attempt in the joint work of Ward and Dilworth. As suggested by their name, principal elements are the analogue in multiplicative lattices of principal ideals in (commutative) rings. Principal elements are the cornerstone on which the theory of multiplicative lattices and abstract ideal theory now largely rests. In this paper, we obtain some new results regarding principal elements and extend some others. In addition, we try to convey what is known and what is not known about the subject. We conclude with a fairly extensive (but likely not exhaustive) bibliography on principal elements.

Generalizations of Multiplicative Ideal Theory to Commutative Rings with Zerodivisors

Generalizations of Multiplicative Ideal Theory to Commutative Rings with Zerodivisors

Multiplicative ideal theory of noncommutative Krull pairs. I: Module systems, Krull ring-type chain conditions, and application to two-sided ideals

Multiplicative ideal theory of noncommutative Krull pairs. I: Module systems, Krull ring-type chain conditions, and application to two-sided ideals

Multiplicative ideal theory of noncommutative Krull pairs. II: Factorization of one-sided ideals

5.1. A Krull pair (E, R) is called strong if every maximal element F in the Asano class 01 of E is d-Noetherian and R is a left-Asano quotient ring of E. This condition entails that every thick divisorial left ideal of every maximal order in iy is symbolically invertible, that is, a member of the Brandt groupoid B&Q) = K (see Theorem 3.6 and Corollary 3.8). In Section 6, the groupoid k’ is investigated from a general point of view. This is used in Section 7 to obtain a factorization theory for one-sided ideals of a strong Krull pair. But since we now have completed the essential definitions, it is appropriate first to point out examples of Krull pairs and strong Krull pairs showing that important classes of rings are of the type considered in this paper.

Multiplicative Ideal Theory. by R. W. Gilmer. Queen's Paper on Pure and Applied Mathematics No. 12. Queen's Univ., Kingston, Ontario (1968). vii+ 700 pp.

Multiplicative Ideal Theory. by R. W. Gilmer. Queen's Paper on Pure and Applied Mathematics No. 12. Queen's Univ., Kingston, Ontario (1968). vii+ 700 pp.

Integral domains in which each non‐zero ideal is divisorial

Let D be an integral domain with identity having quotient field K. A non-zero fractional ideal F of D is said to be divisorial if F is an intersection of principal fractional ideals of D[4; 2]. Equivalently, F is divisorial if there is a non-zero fractional ideal E of D such thatDivisorial ideals arose in the investigations of Van der Waerden, Artin, and Krull in the 1930's and were called v-ideals by Krull [9; 118]. The concept has played an important role in the development of multiplicative ideal theory.

III.—Multiplicative Ideal Theory and Rings of Quotients. I.

SynopsisBy means of a generalized ring of quotients multiplicative ideal theory is studied in an arbitrary (associative) ring. A suitable generalization of the concept of maximal order is given and factorization theorems are obtained for the nonsingular (two sided) ideals, which generalize the theorems of Artin and E. Noether.

On the Derivations in Maximal Orders of Simple Algebras

The theory of derivations and differents in the algebraic number fields or Dedekind rings has been developped by A. Weil [1], Y. Kawada [2] and M. Moriya [4]. Further Y. Kawada [3] has investigated the derivations in maximal orders of simple algebras over number fields. This note is concerned with the derivations in the simple algebras over fields which are quotient fields of Dedekind rings; the commutative rings in which the fundamental theorem of multiplicative ideal theory holds. The auther gives his hearty thanks to Prof. M. Moriya who gave him valuable remarks.