Ore Condition Research Articles

On the Ring of Quotients at a Prime Ideal of a Right Noetherian Ring

J. Lambek and G. Michler [3] have initiated the study of a ring of quotients RP associated with a two-sided prime ideal P in a right noetherian ring R. The ring RP is the quotient ring (in the sense of [1]) associated with the hereditary torsion class τ consisting of all right R-modules M for which HomR(M, ER(R/P)) = 0, where ER(X) is the injective hull of the R-module X.In the present paper, we shall study further the properties of the ring RP. The main results are Theorems 4.3 and 4.6. Theorem 4.3 gives necessary and sufficient conditions for the torsion class associated with P to have property (T), as well as some properties of RP when these conditions are indeed satisfied, while Theorem 4.6 gives necessary and sufficient conditions for R to satisfy the right Ore condition with respect to (P).

Correction to “Localisation in Pi-Rings”

I wish to correct two errors of attribution in “Localisation in PI-rings”, J. London Math. Soc. (2) 2 (1970), 763–768. (a) Theorem 1.1, concerning the Ore condition in stable PI-rings, should have been attributed to L. W. Small (“Localisation in PI-rings”, J. Algebra, 18 (1971), 269–270). (b) The first example in Section 3 should have been attributed to C. Procesi (unpublished) and not, as stated, to L. W. Small. I wish to apologise for any inconvenience caused by these errors.

A Note on Zero Divisors in Group-Rings

Let $Z{G_1}$ and $Z{G_2}$ be the integral group rings of groups ${G_1}$ and ${G_2}$ with a common normal subgroup $H$ and let $K$ be a subgroup of $H$. Let $G$ be the free product of ${G_1}$ and ${G_2}$ amalgamating $K$. If $Z{G_1}$ and $Z{G_2}$ are integral domains and if ZH has the Ore condition then ZG is again an integral domain.

A note on zero divisors in group-rings

Let Z G 1 Z{G_1} and Z G 2 Z{G_2} be the integral group rings of groups G 1 {G_1} and G 2 {G_2} with a common normal subgroup H H and let K K be a subgroup of H H . Let G G be the free product of G 1 {G_1} and G 2 {G_2} amalgamating K K . If Z G 1 Z{G_1} and Z G 2 Z{G_2} are integral domains and if ZH has the Ore condition then ZG is again an integral domain.

Localization and Artinian quotient rings

For a non-commutative ring there exists not always the classical quotient ring, while there exists always the maximal quotient ring. In the case of a semiprime ring, howevel; to have a semi-simple Artinian maximal quotient ring implies the existence of the classical quotient ring (cf. Johnson [6, 4.4 Theorem] and Sandomierski [17, Theorem 1.6]). Therefore avoiding the use of the Ore condition it may be possible to find some conditions, under which the existence of the classical quotient ring is assured by requiring the chain condition on a ring which is constructed by a suitable localization. The purpose of the present paper is to provide such conditions and localizations. For a ring with an identity every idempotent topologizing filter ~is defined by a finitely cogenerating injective right R-module W in the following way: ~ = {right ideal D of R I HomR (R/D, W) = 0}, and we have the localizing functor H such that H(X)=~ Horn R (D, X/X~), where X is a right R-module DE~ and X ~ = {xeXIAnnR x e ~ } . H(R) has a ring-structure and H(X) becomes a right H(R)-module in a natural way. Recently, Morita [14] has shown that H(R) and the double centralizer Q of W R are identical and H is exact. With these notations we can state our main theorem in w 3: The following statements I to III are equivalent.

On categories of quotients

We construct a category of quotients over a category satisfying a condition similar to the Ore condition. Addition of quotients is briefly discussed.

A Counterexample to a Conjecture of D.F. Sanderson

In [2, p. 511] Sanderson has shown that if every Large left ideal of a ring R with identity contains a regular element, and if the regular elements in R satisfy Ore's condition, then the complete (Utumi's) ring of quotients coincides with the classical ring of quotients. He conjectured that the above conditions are also necessary. The following is a counter example.

Reduction of metallic ores by natural gas

In 1960 Ore proved the following theorem: Let G be a graph of order n. If d(u)+d(v)⩾n for every pair of nonadjacent vertices u and v, then G is hamiltonian. In this note we strengthen Ore's theorem as follows: we determine the maximum number of pairs of nonadjacent vertices that can have degree sum less than n (i.e. violate Ore's condition) but still imply that the graph is hamiltonian. Some other sufficient conditions (i.e. Fan's condition) for hamiltonian graphs are strengthened as well.