A Note on Nilpotent Jordan Rings

1987 ◽  
Vol 30 (4) ◽  
pp. 399-401
Author(s):  
Wallace S. Martindale
Keyword(s):  
Associative Ring ◽  
Jordan Ring ◽  
Torsion Free

AbstractLet R be a 2-torsion free associative ring with involution. It is shown that if the set S of symmetric elements is nilpotent as a Jordan ring then R is nilpotent.

Rings with simple Lie rings of Lie and Jordan derivations

2018 ◽  
Vol 17 (04) ◽  
pp. 1850078
Author(s):  
Ahmad Al Khalaf ◽  
Orest D. Artemovych ◽  
Iman Taha
Keyword(s):  
Noetherian Ring ◽  
Associative Ring ◽  
Lie Ring ◽  
Lie Rings ◽  
Lie Derivations ◽  
Torsion Free

Let [Formula: see text] be an associative ring. We characterize rings [Formula: see text] with simple Lie ring [Formula: see text] of all Lie derivations, reduced noncommutative Noetherian ring [Formula: see text] with the simple Lie ring [Formula: see text] of all derivations and obtain some properties of [Formula: see text]-torsion-free rings [Formula: see text] with the simple Lie ring [Formula: see text] of all Jordan derivations.

On Modules of Singular Submodule Zero

1971 ◽  
Vol 23 (2) ◽  
pp. 345-354 ◽  
Author(s):  
Vasily C. Cateforis ◽  
Francis L. Sandomierski
Keyword(s):  
Integral Domain ◽  
Associative Ring ◽  
Quotient Ring ◽  
Essential Extension ◽  
Singular Ideal ◽  
Free Modules ◽  
Torsion Free ◽  
Unitary Right

In this paper we generalize to modules of singular submodule zero over a ring of singular ideal zero some of the results, which are well known for torsion-free modules over a commutative integral domain, e.g. [2, Chapter VII, p. 127], or over a ring, which possesses a classical right quotient ring, e.g. [13, § 5].Let R be an associative ring with 1 and let M be a unitary right R-module, the latter fact denoted by MR. A submodule NR of MR is large in MR (MR is an essential extension of NR) if NR intersects non-trivially every non-zero submodule of MR; the notation NR ⊆′ MR is used for the statement “NR is large in MR” The singular submodule of MR, denoted Z(MR), is then defined to be the set {m ∈ M| r(m) ⊆’ RR}, where

scholarly journals Higher commutators, ideals and cardinality

1996 ◽  
Vol 54 (1) ◽  
pp. 41-54 ◽  
Author(s):  
Charles Lanski
Keyword(s):  
Finite Index ◽  
Associative Ring ◽  
Semiprime Ring ◽  
Torsion Free ◽  
The Ideal

For an associative ring R, we investigate the relation between the cardinality of the commutator [R, R], or of higher commutators such as [[R, R], [R, R]], the cardinality of the ideal it generates, and the index of the centre of R. For example, when R is a semiprime ring, any finite higher commutator generates a finite ideal, and if R is also 2-torsion free then there is a central ideal of R of finite index in R. With the same assumption on R, any infinite higher commutator T generates an ideal of cardinality at most 2cardT and there is a central ideal of R of index at most 2cardT in R.

scholarly journals A Note on Additive Groups of Some Specific Associative Rings

2016 ◽  
Vol 30 (1) ◽  
pp. 219-229
Author(s):  
Mateusz Woronowicz
Keyword(s):  
Associative Ring ◽  
Abelian Groups ◽  
Free Groups ◽  
Additive Group ◽  
Direct Sums ◽  
Rank One ◽  
Elementary Methods ◽  
Associative Rings ◽  
Torsion Free

AbstractAlmost complete description of abelian groups (A, +, 0) such that every associative ring R with the additive group A satisfies the condition: every subgroup of A is an ideal of R, is given. Some new results for SR-groups in the case of associative rings are also achieved. The characterization of abelian torsion-free groups of rank one and their direct sums which are not nil-groups is complemented using only elementary methods.

scholarly journals The Brown Mccoy radical of semigroup rings of commutative cancellative semigroups

1985 ◽  
Vol 26 (2) ◽  
pp. 107-113 ◽  
Author(s):  
E. Jespers ◽  
J. Krempa ◽  
P. Wauters
Keyword(s):  
Quotient Group ◽  
Associative Ring ◽  
Semigroup Ring ◽  
Cancellative Semigroup ◽  
Semigroup Rings ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free

We give a complete description of the Brown–McCoy radical of a semigroup ring R[S], where R is an arbitrary associative ring and S is a commutative cancellative semigroup; in particular we obtain the answer to a question of E. Puczyłowski stated in [11]Throughout this note all rings R are associative with unity 1; all semigroups S are commutative and cancellative with unity. Note that the condition that R and S have a unity can be dropped (cf. [8]). The quotient group of S is denoted by Q(S). We say that S is torsion free (resp. has torsion free rank n) if Q(S) is torsion free (resp. has torsion free rank n). The Brown–McCoy radical (i.e. the upper radical determined by the class of all simple rings with unity) of a ring R is denoted by u(R). We refer to [2] for further detail on radicals and in particular on the Brown–McCoy radical.First we state some well-known results and a preliminary lemma. Let R and T be rings with the same unity such that R ⊂ T. Then T is said to be a normalizing extension of R if T = Rx1+…+Rxn for certain elements x1, …, xn of T and Rxi = xiR for all i such that 1 ∨i∨n. If all xi are central in T, then we say that T is a central normalizing extension of R.

scholarly journals The type set of a torsion-free abelian group of rank two

1979 ◽  
Vol 27 (4) ◽  
pp. 507-510 ◽  
Author(s):  
David R. Jackett
Keyword(s):  
Abelian Group ◽  
Recent Result ◽  
Associative Ring ◽  
Free Abelian Group ◽  
Torsion Free Abelian Group ◽  
Torsion Free

AbstractIn this paper we generalize a recent result of Freedman (1973) concerning the cardinality of the type set of a rank two torsion-free abelian group. We show that if A is such a group and A supports a non-trivial associative ring then the type set of A contains at most three elements.

scholarly journals On the automorphisms of the group ring of a finitely generated free abelian group

1988 ◽  
Vol 31 (1) ◽  
pp. 71-75
Author(s):  
M. M. Parmenter
Keyword(s):  
Abelian Group ◽  
Prime Ideal ◽  
Group Ring ◽  
Associative Ring ◽  
Free Abelian Group ◽  
Finitely Generated ◽  
Torsion Free Abelian Group ◽  
Special Case ◽  
Torsion Free

Let R be an associative ring with 1 and G a finitely generated torsion-free abelian group. In this note, we classify all R-automorphisms of the group ring RG. The special case where G is infinite cyclic was previously settled in [8], and our interest in this problem was rekindled by the recent paper of Mehrvarz and Wallace [7], who carried out the classification in the case where R contains a nilpotent prime ideal.

E-Associative Rings

1993 ◽  
Vol 36 (2) ◽  
pp. 147-153 ◽  
Author(s):  
Shalom Feigelstock
Keyword(s):  
Associative Ring ◽  
Additive Group ◽  
Associative Rings ◽  
Torsion Free

AbstractA ring R is E-associative if φ(xy) = φ(x)y for all endomorphisms φ of the additive group of R, and all x,y ∊ R. Unital E-associative rings are E-rings. The structure of the torsion ideal of an E-associative ring is described completely. The E-associative rings with completely decomposable torsion free additive groups are also classified. Conditions under which E-associative rings are E-rings, and other miscellaneous results are obtained.

scholarly journals Commutativity theorems for rings and groups with constraints on commutators

1984 ◽  
Vol 7 (3) ◽  
pp. 513-517 ◽  
Author(s):  
Evagelos Psomopoulos
Keyword(s):  
Associative Ring ◽  
Commutativity Theorems ◽  
Torsion Freeness ◽  
Positive Integers ◽  
Torsion Free

Letn>1,m,t,sbe any positive integers, and letRbe an associative ring with identity. Supposext[xn,y]=[x,ym]ysfor allx,yinR. If, further,Risn-torsion free, thenRis commutativite. Ifn-torsion freeness ofRis replaced by “m,nare relatively prime,” thenRis still commutative. Moreover, example is given to show that the group theoretic analogue of this theorem is not true in general. However, it is true whent=s=0andm=n+1.

scholarly journals Commutativity results for rings

1988 ◽  
Vol 38 (2) ◽  
pp. 191-195 ◽  
Author(s):  
Hazar Abu-Khuzam
Keyword(s):  
Positive Integer ◽  
Finite Subset ◽  
Associative Ring ◽  
Free Ring ◽  
Commutator Ideal ◽  
Fixed Positive Integer ◽  
Torsion Free

Let R be an associative ring. We prove that if for each finite subset F of R there exists a positive integer n = n(F) such that (xy)n − yn xn is in the centre of R for every x, y in F, then the commutator ideal of R is nil. We also prove that if n is a fixed positive integer and R is an n(n + 1)-torsion-free ring with identity such that (xy)n − ynxn = (yx)n xnyn is in the centre of R for all x, y in R, then R is commutative.

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