Chan Huh

ARMENDARIZ RINGS AND SEMICOMMUTATIVE RINGS

In this note we concern the structures of Armendariz rings and semicommutative rings which are generalizations of reduced rings, the classical right quotient rings of Armendariz rings, the polynomial rings over semicommutative rings, and the relationships between Armendariz rings and semicommutative rings. We actually show that (i) for a right Ore ring R with Q its classical right quotient ring, R is Armendariz if and only if Q is Armendariz; (ii) for a semiprime right Goldie ring R with Q its classical right quotient ring, R is Armendariz , R is reduced , R is semicommutative , Q is Armendariz , Q is reduced , Q is semicommutative , Q is a finite direct product of division rings; (iii) there is a semicommutative ring over which the polynomial ring need not be semicommutative; and (iv) Armendariz rings need not be semicommutative. Moreover we extend the classes of Armendariz rings and semicommutative rings, observing the conditions under which some kinds of rings may be Armendariz or semicommutative.

p.p. rings and generalized p.p. rings

Abstract This paper concerns two conditions, called right p.p. and generalized right p.p., which are generalizations of Baer rings and von Neumann regular rings. We study the subrings and extensions of them, adding proper examples and counterexamples to some situations and questions that occur naturally in the process of this paper.

NILRADICALS OF POWER SERIES RINGS AND NIL POWER SERIES RINGS

Klein proved that polynomial rings over nil rings of bounded index are also nil of bounded index; while Puczylowski and Smoktunowicz described the nilradical of a power series ring with an indeterminate. We extend these results to those with any set of commuting indeterminates. We also study prime radicals of power series rings over some class of rings containing the case of bounded index, flnding some examples which elaborate our arguments; and we prove that R is a PI ring of bounded index then the power series ring R((X)), with X any set of indeterminates over R, is also a PI ring of bounded index, obtaining the Kleins result for polynomial rings as a corollary.

RING WHOSE MAXIMAL ONE-SIDED IDEALS ARE TWO-SIDED

In this note we are concerned with relationships between one-sided ideals and two-sided ideals, and study the properties of polynomial rings whose maximal one-sided ideals are two-sided, in the viewpoint of the Nullstellensatz on noncommutative rings. Let R be a ring and R[x] be the polynomial ring over R with x the indeterminate. We show that eRe is right quasi-duo for if R is right quasi-duo; R/J(R) is commutative with J(R) the Jacobson radical of R if R[] is right quasi-duo, from which we may characterize polynomial rings whose maximal one-sided ideals are two-sided; if R[x] is right quasi-duo then the Jacobson radical of R[x] is N(R)[x] and so the conjecture (i.e., the upper nilradical contains every nil left ideal) holds, where N(R) is the set of all nilpotent elements in R. Next we prove that if the polynomial rins R[x], over a reduced ring R with X 2, is right quasi-duo, then R is commutative. Several counterexamples are included for the situations that occur naturally in the process of this note.

On rings in which every maximal one-sided ideal contains a maximal ideal

Given a ring R, consider the condition: (*) every maximal right ideal of R contains a maximal ideal of R. We show that, for a ring R and 0 ≠ e 2 = e ∈ R such that ele ⫋ eRe every proper ideal I of R R satisfies (*) if and only if eRe satisfies (*). Hence with the help of some other results, (*) is a Morita invariant property. For a simple ring R R[x] satisfies (*) if and only if R[x] is not right primitive. By this result, if R is a division ring and R[x] satisfies (*), then the Jacobson conjecture holds. We also show that for a finite centralizing extension S of a ring R R satisfies (*) if and only if S satisfies (*).

On a Hereditary Radical Property Relating to the Reducedness

In this note, a hereditary radical property, called homomorphically reduced rings, is introduced, observed, and applied. The dual concept of this property is also studied with the help of Courter, proving that any ring R (possibly without identity) has an ideal S such that S/K is not homomorphically reduced for each proper ideal K of S; and if L is an ideal of R with L ⊊ S, then L/H is homomorphically reduced for some ideal H of R with H ⊊ L. The concept of the homomorphical reducedness is shown to be equivalent to the left (right) weak regularity and the (strong) regularity for one-sided duo rings. It is proved that homomorphically reduced rings have several useful properties similar to those of (weakly) regular rings. It is proved that the homomorphical reducedness can go up to classical quotient rings. It is shown that if R is a reduced right Ore ring with the ascending chain condition (ACC) for annihilator ideals, then the maximal right quotient ring of R is strongly regular (hence homomorphically reduced).

AN ANDERSON'S THEOREM ON NONCOMMUTATIVE RINGS

Abstract. Let R be a ring and I be a proper ideal of R . For the caseof R being commutative, Anderson proved that ( ∗ ) there are only finitelymany prime ideals minimal over I whenever every prime ideal minimalover I is finitely generated. We in this note extend the class of ringsthat satisfies the condition ( ∗ ) to noncommutative rings, so called ho-momorphically IFP , which is a generalization of commutative rings. Asa corollary we obtain that there are only finitely many minimal primeideals in the polynomial ring over R when every minimal prime ideal ofa homomorphically IFP ring R is finitely generated. Throughout every ring is associative with identity unless otherwise stated.The n by n matrix ring over a ring R is denoted by Mat n ( R ). Due to Bell [2], aright (or left) ideal I of a ring R is said to have the insertion-of-factors-property (simply IFP ) if ab ∈ I implies aRb ⊆ I for a,b ∈ R . A ring R is called IFP ifthe zero ideal of R has the IFP. For a ring R and an ideal I , note that

On a ring structure related to annihilators

In this note, we focus our attention on a new ring structure related to annihilators, and consider a ring property that contains many kinds of ring classes, introducing right ZAFS. This property is shown to be not left-right symmetric but left-right symmetric for left or right Artinian rings. The left (right) ZAFS property is shown to pass to Ore extensions with automorphisms. The left (respectively, right) ZAFS property is shown to pass also to classical left (respectively, right) quotient rings, yielding that semiprime right Goldie rings are ZAFS.