Yasuyuki Hirano

On annihilator ideals of a polynomial ring over a noncommutative ring

Abstract Let R be a ring and let R[x] denote the polynomial ring over R. We study relations between the set of annihilators in R and the set of annihilators in R[x].

ON ORDERED MONOID RINGS OVER A QUASI-BAER RING

A ring R is called (left principally) quasi-Baer if the left annihilator of every (principal) left ideal of R is generated by an idempotent. We show that if R is (left principally) quasi-Baer and G is an ordered monoid, then the monoid ring RG is again (left principally) quasi-Baer. When R is (left principally) quasi-Baer and G is an ordered group acting on R, we give a necessary and sufficient condition for the skew group ring R♯G to be (left principally) quasi-Baer.

On strongly bounded rings and duo rings

A ring R is said to be strongly right (left) bounded if every nonzero right (left) ideal contains a nonzero ideal; and R is right bounded if every essential right ideal contains a nonzero ideal. Strongly right bounded rings play a fundamental role in the theory of FPF rings (e.g., a strongly right bounded

Rings in which every element is the sum of two idempotents

Let R be a ring with prime radical P . The main theorems of this paper are (1) The following conditions are equivalent.: 1) R is a commutative ring in which every element is the sum of two idempotents; 2) R is a ring in which every element is the sum of two commuting idempotents; 3) R satisfies the identity x 3 = x . (2) If R is a PI-ring in which every element is the sum of two idempotents, then R/P satisfies the identity x 3 = x . (3) Let R be a semi-perfect ring in which every element is the sum of two idempotents. If R R R is quasi-projective, then R is a finite direct sum of copies of GF (2) and/or GF (3).

ON RINGS IN WHICH EVERY IDEAL IS WEAKLY PRIME

Anderson-Smith (1) studied weakly prime ideals for a com- mutative ring with identity. Blair-Tsutsui (2) studied the structure of a ring in which every ideal is prime. In this paper we investigate the struc- ture of rings, not necessarily commutative, in which all ideals are weakly prime. condition for a ring to have such property is given and several examples to support given propositions are constructed. We then further investigate commutative rings in which every ideal is weakly prime and the structure of such rings under assumptions that generalize commutativity of rings. At the end, we consider the structure of rings in which every right ideal is weakly prime. Definitions and general results

On a problem of Szász

Let R be a ring with centre Z . In this note, we prove the following: If the additive group Z + of Z has finite group-theoretic index in R + , then R has an ideal I contained in Z such that R/I is a finite ring. This is a solution of a problem posed by F.A. Szasz.

ON A THEOREM OF AYAD AND RYCKELYNCK

Abstract Let d be a K-derivation of the polynomial ring K[x 1 , …, x n ] over a field K of characteristic 0, and let ᵭ be the extension of d to the fraction field K(x 1 ,…, x n ). Recently Ayad and Ryckelynck proved that if the kernel Ker d of d contains n − 1 algebraically independent polynomials then Ker ᵭ is equal to the fraction field Q(Ker d) of Ker d. In this note, we give a short proof for this result.

Splitting Superhereditary Preradicals

Abstract We study the structure of rings R with the property that, for every right R-module M and an ideal I of R, the annihilator of I in M is a direct summand of M.

A GENERALIZATION OF SEMISIMPLE ARTINIAN RINGS

We investigate a ring R with the property that for every right R-module M and every ideal I of R the annihilator of I in M is a direct summand of M, and determine conditions under which such a ring is semisimple Artinian.

On pairs of subrings with a common set of proper ideals

A study of pairs of commutative rings with the same set of prime ideals appears in the literature. In this paper, we investigate pairs of subrings, not necessarily commutative, with a common set of proper ideals.