Symmetric property of rings with respect to the Jacobson radical

2016

Let R be a ring with identity and J(R) denote the Jacobson radical of R. A ring R is called J-reduced if a = 0 (n ∈ N) implies a ∈ J(R) for any a ∈ R. We give, in this article, many characterizations of such rings. We construct some families of J-reduced rings. Furthermore, the transposes of invertible matrices over such rings are studied.

2018

We introduce a weakly symmetric ring which is a generalization of a symmetric ring and a strengthening of both a GWS ring and a weakly reversible ring, and investigate properties of the class of this kind of rings. A ring R is called weakly symmetric if for any a, b, c ∈ R, abc being nilpotent implies that Racrb is a nil left ideal of R for each r ∈ R. Examples are given to show that weakly symmetric rings need to be neither semicommutative nor symmetric. It is proved that the class of weakly symmetric rings lies also between those of 2-primal rings and directly finite rings. We show that for a nil ideal I of a ring R, R is weakly symmetric if and only if R/I is weakly symmetric. If R[x] is weakly symmetric, then R is weakly symmetric, and R[x] is weakly symmetric if and only if R[x;x−1] is weakly symmetric. We prove that a weakly symmetric ring which satisfies Köthe’s conjecture is exactly an NI ring. We also deal with some extensions of weakly symmetric rings such as a Nagata exte...

Canadian Mathematical Bulletin

This paper is about rings $R$ for which every element is a sum of a tripotent and an element from the Jacobson radical $J(R)$ . These rings are called semi-tripotent rings. Examples include Boolean rings, strongly nil-clean rings, strongly 2-nil-clean rings, and semi-boolean rings. Here, many characterizations of semi-tripotent rings are obtained. Necessary and sufficient conditions for a Morita context (respectively, for a group ring of an abelian group or a locally finite nilpotent group) to be semi-tripotent are proved.

2008

A J-ring is a ring R with the property that for every x in R there exists an integer n(x)>1 such that x x x n = ) ( , and a well-known theorem of Jacobson states that a Jring is necessarily commutative. With this as motivation, we define a generalized Jring to be a ring R with the property that for all x, y in R0 there exists integers 1

Pacific Journal of Mathematics, 1973

Journal of Mathematical Sciences, 2008

Communications in Algebra, 2001

All commutative semigroups S are described such that the Jacobson radical is homogeneous in each ring graded by S.

Transactions of the American Mathematical Society, 1971

If R is an associative ring, we consider the special Jordan ring R + {R^ + } , and when R has an involution, the special Jordan ring S of symmetric elements. We first show that the prime radical of R equals the prime radical of R + {R^ + } , and that the prime radical of R intersected with S is the prime radical of S. Next we give an elementary characterization, in terms of the associative structure of R, of primeness of S. Finally, we show that a prime ideal of R intersected with S is a prime Jordan ideal of S.

Journal of Algebra, 1981

For example, an associative algebra A gives rise to a Jordan algebra .Z = A9 by taking U, y = xyx and x2 = xx. A Jordan algebra is unitaf if there is an element 1 with U,x = x and U, 1 =x2 for all x. Any Jordan algebra can be imbedded in a unital Jordan algebra, its unital hull J' = Qi 1 + J. An element u of a unital Jordan algebra is invertible if U,, is an invertible operator, which is equivalent to 1 E U,J. An element x of J is quasi-invertible if 1-x is invertible in J', i.e., if U ,-+ = Z-V, + U, is invertible, equivalently if U,-,J= J. An element is properly quasi-invertible if it remains quasi-invertible in all homotopes JCy' of J, where the operations in the homotope are given by x is quasi-invertible in Jfy' iff U&, = 1-VX*y + U, U,, = TX,y is invertible (on J or J'), equivalently iff T,,,(J) = J. The Jacobson radical of a Jordan algebra is the maximal ideal of quasiinvertible elements, and consists precisely of all properly quasi-invertible elements [3]. We wish to relate this to inner ideals. We define a weak inner ideal to be a subspace I,, c J invariant under inner multiplication by J, Ur,J c I,, while a (strict) inner ideal Z is a weak inner ideal which remains such in J', UtJ' c I. Thus an inner ideal satisfies the additional condition that U, 1 = Z2 c I. (In a unital algebra all weak inner ideals are inner). Formula (0.2) shows U,J is an inner ideal in J for any y E .Z', and (0.3) shows T,,,J is an inner ideal in J for any x, y E J'. If J = A9 for an associative algebra A, then any right or left ideal B in A is an inner ideal in J.

Arabian Journal of Mathematics

The class of J-normal rings lies between the classes of weakly normal rings and left min-abel rings. It is proved that R is J-normal if and only if for any idempotent e ∈ R and for any r ∈ R, R(1 − e)re ⊆ J (R) if and only if for any n ≥ 1, the n × n upper triangular matrix ring U n (R) is a J-normal ring if and only if the Dorroh extension of R by Z is J-normal. We show that R is strongly regular if and only if R is J-normal and von Neumann regular. For a J-normal ring R, it is obtained that R is clean if and only if R is exchange. We also investigate J-normality of certain subrings of the ring of 2 × 2 matrices over R.