The prime radical in special Jordan rings

Bulletin of The Australian Mathematical Society, 1993

Transactions of the American Mathematical Society, 1971

In this paper a definition is given for a prime ideal in an arbitrary nonassociative ring N under the single restriction that for a given positive integer iä2, if A is an ideal in N, then A" is also an ideal. (N is called an i-naring.) This definition is used in two ways. First it is used to define the prime radical of N and the usual theorems ensue. Second, under the assumption that the s-naring N has a certain property (a), the Levitzki radical L(N) of N is defined and it is proved that L(N) is the intersection of those prime ideals P in N whose quotient rings are Levitzki semisimple. N has property (a) if and only if for each finitely generated subring A and each positive integer m, there is an integer/(m) such that Af(m,^Am. (Here Ai = As and Am + 1 = A'm.) Furthermore, conditions are given on the identities an s-naring N satisfies which will insure that N satisfies (a). It is then shown that alternative rings, Jordan rings, and standard rings satisfy these conditions.

Communications of The Korean Mathematical Society, 2019

Let R be a ring with identity and J(R) denote the Jacobson radical of R, i.e., the intersection of all maximal left ideals of R. A ring R is called J-symmetric if for any a, b, c ∈ R, abc = 0 implies bac ∈ J(R). We prove that some results of symmetric rings can be extended to the Jsymmetric rings for this general setting. We give many characterizations of such rings. We show that the class of J-symmetric rings lies strictly between the class of symmetric rings and the class of directly finite rings.

Let R be a ring and S a nonempty subset of R. Suppose that θ and ϕ are endomorphisms of R. An additive mapping δ:R→R is called a left (θ,ϕ)-derivation (resp., Jordan left (θ,ϕ)-derivation) on S if δ(xy)=θ(x)δ(y)+ϕ(y)δ(x) (resp., δ(x2)=θ(x)δ(x)+ϕ(x)δ(x)) holds for all x,y∈S. Suppose that J is a Jordan ideal and a subring of a 2-torsion-free prime ring R. In the present paper, it is shown that if θ is an automorphism of R such that δ(x2)=2θ(x)δ(x) holds for all x∈J, then either J⫅Z(R) or δ(J)=(0). Further, a study of left (θ,θ)-derivations of a prime ring R has been made which acts either as a homomorphism or as an antihomomorphism of the ring R.

Proceedings of the American Mathematical Society, 1986

There are important connections between radicals of a special Jordan algebra J J and its associative envelope A A . For the locally nilpotent (Levitzki) radical L \mathcal {L} , Skosyrskii proved L ( J ) = J ∩ L ( A ) \mathcal {L}(J) = J \cap \mathcal {L}(A) . For the prime (Baer) radical P \mathcal {P} , Erickson and Montgomery proved P ( J ) = J ∩ P ( A ) \mathcal {P}(J) = J \cap \mathcal {P}(A) when J = H ( A , ∗ ) J = H(A, * ) consists of all symmetric elements of an algebra A A with involution ∗ * . In his important work on prime Jordan algebras, Zelmanov proved P ( J ) = J ∩ P ( A ) \mathcal {P}(J) = J \cap \mathcal {P}(A) for all linear J J and all associative envelopes A A . In the present paper we extend Zelmanov’s result to arbitrary quadratic Jordan algebras. In particular, we see that a special Jordan algebra is semiprime iff it has some semiprime associative envelope.

Indian Journal of Science and Technology, 2020

Objectives: The main objective of this article is to introduce *-Jordan ideals of a certain class of semirings called MA-semirings with involution and to investigate some conditions for which the above said ideals contained in the center. Methods and findings: We use the Jacobian identities and 2-torsion freeness of MA semirings. In this connection, we establish some important results of ring theory for the class of MA-semirings. Applications/ improvements: The commutative property is helpful to study the theory of semirings with ease therefore we find some conditions to impose commutativity in semirings, which are indeed novel idea in the field of semirings. Furthermore, these conditions are used in a most generalized way that these conditions bring the *-Jordan ideals to the center, therefore, it would be the corollary of result that semiring is commutative.

Communications in Algebra, 2000

Strongly prime rings may be defined as prime rings with simple central closure. This paper is concerned with further investigation of such rings. Various characterizations, particularly in terms of symmetric zero divisors, are given. We prove that the central closure of a strongly (semi-)prime ring may be obtained by a certain symmetric perfect one sided localization. Complements of strongly prime ideals are described in terms of strongly multiplicative sets of rings. Moreover, some relations between a ring and its multiplication ring are examined.

Publicationes Mathematicae Debrecen

Special radicals were defined for rings with involution by Salavov a. In this paper we show that every special radical R in the variety of rings induces a corresponding special radical R * in the variety of rings with involution, and R * (R) ⊆ R(R) for any involution ring R. The reverse inclusion does not hold in general. This theory gives new characterisations for certain concrete radicals.

Journal of Algebra, 1974

1968

Historically the radical has been based on the notion of nilpotency. This study has only yielded significant results in rings with chain conditions. In 1945 N. Jacobson defined the first radical that gave significant results without chain conditions. Since that time about eight radicals have been defined and investigated for arbitrary rings.' The purpose of this report is to investigate two of these radicals and study certain properties of these radicals. The two radicals considered are the prime radical and the Jacobson radical. Henceforth in this paper the word radical will refer to one of these two radicals. These radicals are of interest since they are different from the classical radical which was defined as the maximal nilpotent ideal, but coincide with it in rings with the descending chain condition. The prime radical is first defined in three ways. Then using the concept of an msystem the definitions are shown to be equivalent. In the process properties of prime and semi-prime ideals are discussed. Several properties of the prime radical are also proved including the fact that it is a nil ideal containing all nilpotent ideals. This fact is important in the relati between the prime radical and the Jacobson radical. Rings related to a ring R and the results for the prime radical are discussed. The related rings on considered are ideals in a ring R, homomorphic images of R and the ring of n by n matrices over R. In a similar form the Jacobson radical is discussed. Again three definitions are considered. Starting with the concept of quasi-regularity these are shown to be equivalent. Several properties of quasi-regularity and the Jacobson radical are proved. Also the concepts of prin:iitive ring and R-modulc are used. These are understood to n:iean right primitive and right R-module in the discussion of the Jacobson radical. Then the radical in the related rings are considered, that is the ideals of a ring R, homomorphic images of a ring R and the complete matrix ring of a ring R. The final section considers rings with the descending chain condition. First the containment of the prime radical in the Jacobson radical is shown. Then an example of the distinctness of the radicals is given. Next a theorem is proved that establishes that in the rings with descending chain condition the Jacobson radical is nilpotent. This result together with others in the report yields that in rings with descending chain condition the prime radical and Jacobson radical coincide. In this report a ring need not have an identity element unless so specified. An ideal will be two-sided unless modified by right or left. A list of conventions and notation used is found in the Appendix of this paper, roughly in the order in which they appear.

Journal of the Australian Mathematical Society, 1995

Let λ be a property that a lattice of submodules of a module may possess and which is preserved under taking sublattices and isomorphic images of such lattices and is satisfied by the lattice of subgroups of the group of integer numbers. For a ring R the lower radical Λ generated by the class λ(R) of R-modules whose lattice of submodules possesses the property λ is considered. This radical determines the unique ideal Λ (R) of R, called the λ-radical of R. We show that Λ is a Hoehnke radical of rings. Although generally Λ is not a Kurosh-Amitsur radical, it has the ADS-property and the class of Λ-radical rings is closed under extensions. We prove that Λ (Mn (R)) ⊆ Mn (Λ(R)) and give some illustrative examples.

Reverse and Jordan ( , ) − biderivation on Prime and Semi-prime Rings, 2019

In this study, we prove that any nonzero reverse (,) − biderivation on a prime ring is (,) − biderivation. Also, we show that any Jordan (,) − biderivation on non-commutative semi-prime ring with ℎ () ≠ 2 is an (,) − biderivation. In addition, we investigate commutative feature of prime ring with Jordan left (,) − biderivation.

International Journal of Computer Applications, 2017

In this paper we prove some properties of ideals that are preserved under localization. Also, we establish several one to one correspondences between certain types of ideals in the ring and its localization at multiplicative systems. We introduce two concepts such as prime radical and minimal prime ideals and the relations of prime radical with the prime radical and Jacobson radical of a ring under localization are determined. Also, we prove a one to one correspondence between minimal primal ideals of a given ideal of a ring and the minimal prime ideals of the ideal of , where is a prime ideal of .

Proceedings of the National Academy of …, 1966

Pacific Journal of Mathematics, 1975

We give a systematic account on the relationship between a ring R with involution and its subrings S and K, which are generated by all its symmetric elements or skew elements respectively. I. Introduction. Let R be a ring with involution * and 5 the subring generated by the set S of all symmetric elements in R. The relationship between R and S has been studied by various authors. In [3] Dieudonne showed that if R is a division ring of characteristic not 2, then either S = R or SQZ(R), the center of R. Later Herstein [4] extended this result by proving S = R for any simple ring R with dim z i?>4 and char.i?^2. The restriction on characteristic was removed by Montgomery [12]. Recently, Lanski [9] proved that if_R is prime or semi-prime, so is 5. In §2 of this paper, we show that S can inherit a number of ring-theoretic properties such as primitivity, semisimplicity, absence of nonzero nil ideals etc.. In doing so, a notion called symmetric subring, which is a generalization of S and its *homomorphic images, is introduced so that a group of theorems of the same type, including Lanski's results, can be proved via a more or less unified argument. We show also that numerous radicals of S are merely the contractions from those of R. As a consequence, we see that R modulo its prime radical behaves much like S in many respects. In §3 we establish a corresponding theory for K, the subring generated by all skew elements. The only result hitherto known concerning K was as follows [4], [12]: If R is_simple and dim z i? >4, then K-R L As a matter of fact, the subring K 2 is more closely related to JR than K is. We apply thejtechnique developed in §2 to study the relationship between R and K 2 , and then derive some parallel theorems for K. II. Symmetric subrings. Our work depends heavily on the notion of Lie ideals. By a Lie ideal U of R we mean an additive subgroup which is invariant under all inner derivations of JR. That is, [u y x] = ux-xu E U for all u E U and x E R. The following lemma concerning Lie ideals will be referred to frequently in the sequel, and it is a combination of some results in [5].