Abstract. We show that the θ-prime radical of a ring R is the set of all strongly θ-nilpotent elements in R, where θ is an automorphism of R. We observe some conditions under which the θ-prime radical of R coincides with the prime radical of R. Moreover we characterize elements in prime radicals of skew Laurent polynomial rings, studying (θ, θ −1 )-(semi)primeness of ideals of R.

Abstract. We first show that the semiprimeness, primeness, and reducedness can go up to up-monoid rings. By these results we can compute the lower nilradicals of up-monoid rings, from which the well-known fact of Amitsur and McCoy for the polynomial rings can be extended to up-monoid rings.A monoid G is called a unique product monoid (simply, up-monoid) if given any two nonempty finite subsets A and B of G there exists at least one c ∈ G that has a unique representation in the form c = ab with a ∈ A and b ∈ B. A group is called a up-group if it satisfies the preceding condition. The study of up-monoids has important roles in group theory and ring theory (see [6], [7] for more details). Group algebras of up-groups are extensively observed relating to the zero divisor problem (see [7]). These lead us to study the basic structure of monoid rings of up-monoids relating to the (semi)primeness and reducedness. Many other relevant results can be found in [1] and [2].Throughout this note each ring is associative and possibly without identity. A ring is called reduced if it has no nonzero nilpotent elements. A ring is called semiprime if the prime radical is zero. Reduced rings are clearly semiprime and note that a commutative ring is semiprime if and only if it is reduced.Let R be a reduced ring. Then with the help of [5] we have that if

Abstract. We continue the study of McCoy condition to analyze zerodividing polynomials for the constant annihilators in the ideals generated by the coefficients. In the process we introduce the concept of ideal-π-McCoy rings, extending known results related to McCoy condition. It is shown that the class of ideal-π-McCoy rings contains both strongly McCoy rings whose non-regular polynomials are nilpotent and 2-primal rings. We also investigate relations between the ideal-π-McCoy property and other standard ring theoretic properties. Moreover we extend the class of ideal-π-McCoy rings by examining various sorts of ordinary ring extensions. Ideal-π-McCoy ringsThroughout this note every ring is associative with identity unless otherwise stated. Let R be a ring and we use R[x] to denote the polynomial ring with an indeterminate x over R. Denote the n by n full matrix ring over R by Mat n (R) and the n by n upper (resp. lower) triangular matrix ring over R by U n (R) (resp. L n (R)). Use E ij for the matrix with (i, j)-entry 1 and elsewhere 0. Z and Z n denote the set of integers and the ring of integers modulo n, respectively.). We will apply these isomorphisms freely. N * (R) and N (R) denote the prime radical and the set of all nilpotent elements in R, respectively.

The concept of reflexive property is introduced by Mason. This note concerns a ring-theoretic property of matrix rings over reflexive rings. We introduce the concept of weakly reflexive rings as a generalization of reflexive rings. From any ring, we can construct weakly reflexive rings but not reflexive, using its lower nilradical. We study various useful properties of such rings in relation with ideals in matrix rings, showing that this new property is Morita invariant. We also investigate the weakly reflexive property of several sorts of ring extensions which have roles in ring theory.

This paper, concerns a class of rings which satisfies the Abelian property in relation to the insertion property at zero by powers and local finite. The concepts of Insertion of-Power-Factors-Property (PFP) and principal finite are introduced for the purpose, and the structures of IPFP, Abelian and locally (principally) finite rings are investigated in relation with several situations of matrix rings and polynomial rings. Moreover, the results obtained here are widely applied to various sorts of rings which have roles in the noncommutative ring theory.