The study of images of noncommutative polynomials on algebras has attracted considerable attention. We investigate polynomial images and the additive structures they generate in associative algebras, focusing on sums and products of values. Motivated by results on additive commutators, we show that finite sums of such products on a nonzero ideal must contains a nonzero ideal, with only minor exceptions. Consequently, for a simple algebra, the subring generated by the image of a noncentral polynomial coincides with the whole algebra, up to a small exceptional case. We further study representations of elements as sums of products of polynomial values, and examine products of additive commutators for matrices over division rings. To simplify multilinear polynomials, we introduce decomposable polynomials and show that, in many cases, their images equal the whole algebra. Finally, we consider polynomial commutators and prove that every noncommutative infinite simple algebra is generated by such elements, together with results on multiplicative commutators, including a complete description for real quaternions.

A prime ring R with extended centroid C is said to be exceptional if both charR = 2 and dimCRC = 4. Herstein characterized additive subgroups A of a nonexceptional simple ring R satisfying [A, [R,R]] subset of A. In 1972, Lanski and Montgomery extended Herstein's theorem to nonexceptional prime rings. In the paper, we first extend Herstein's theorem to arbitrary simple rings. For the prime case, let R be an exceptional prime ring with center Z(R). It is proved that if A is a noncentral additive subgroup of R satisfying [A,L] subset of A for some nonabelian Lie ideal L of R, then beta Z(R) subset of A for some nonzero beta is an element of Z(R), and either AC = Ca + C for some a is an element of A\Z(R) with a2 is an element of Z(R) or [RC,RC] subset of AC. Second, we study certain generalized linear identities satisfied by Lie ideals and then completely characterize derivations delta,d of R satisfying delta d(L) subset of Z(R) for L a Lie ideal of R.

Motivated by some recent results on Lie ideals, it is proved that if L is a Lie ideal of a simple or [R, R] C_ L, which gives a generalization of a classical theorem due to Herstein. We also study commutators and products of noncentral Lie ideals of prime rings. Precisely, let R be a prime ring with extended centroid C. We completely characterize Lie ideals L and elements a of R such that L + aL contains a nonzero ideal of R. Given noncentral Lie ideals K, L of R, it is proved that [K, L] = 0 if and only if KC = LC = Ca + C for any noncentral element a E L. As a consequence, we characterize noncentral Lie ideals K1, ... , Km with m >= 2 such that K1K2 Km contains a nonzero ideal of R. Finally, we characterize noncentral Lie ideals Kj's and Lk's satisfying [K1K2 Km, L1L2 Ln] = 0 from the viewpoint of centralizers. (c) 2025 Elsevier GmbH. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Let R be a prime ring with center Z(R) and with involution & lowast;. Given an additive subgroup A of R, let T(A) := {x + x(& lowast;)|x is an element of A} and K-0(A) := {x - x(& lowast;)|x is an element of A}. Let L be a non-abelian Lie ideal of R. It is proved that if d is a nonzero derivation of R satisfying d(T(L)) = 0 (respectively, d(K-0(L)) = 0), then T(R)(2) subset of Z(R) (respectively, K-0(R)(2) subset of Z(R)). These results are applied to the study of d(T(M)) = 0 and d(K-0(M)) = 0 for non-central & lowast;-subrings M of a division ring R such that M is invariant under all inner automorphisms of R, and for non-central additive subgroups M of a prime ring R containing a nontrivial idempotent such that M is invariant under all special inner automorphisms of R. The obtained theorems also generalize some recent results on simple Artinian rings with involution due to Chacron.

We prove conditions ensuring that a Lie ideal or an invariant additive subgroup in a ring contains all additive commutators. A crucial assumption is that the subgroup is fully noncentral, that is, its image in every quotient is noncentral. For a unital algebra over a field of characteristic where every additive commutator is a sum of square‐zero elements, we show that a fully noncentral subspace is a Lie ideal if and only if it is invariant under all inner automorphisms. This applies in particular to zero‐product balanced algebras.

We study the functional identity G(x)f(x)=H(x) on a division ring D, where f:D→D is an additive map and G(X)≠0,H(X) are generalized polynomials in the variable X with coefficients in D. Precisely, it is proved that either D is finite-dimensional over its center or f is an elementary operator. Applying the result and its consequences, we prove that if D is a noncommutative division ring of characteristic not 2, then the only solution of additive maps f,g on D satisfying the identity f(x)=xng(x−1) with n≠2 a positive integer is the trivial case, that is, f=0 and g=0. This extends Catalano and Merchán's result in 2023 to get a complete solution.

For a nonempty subset X of a ring R, the ring R is called X-semiprime if, given a is an element of R, aXa = 0 implies a = 0. This provides a proper class of semiprime rings. First, we clarify the relationship between idempotent semiprime and unit-semiprime rings. Secondly, given a Lie ideal L of a ring R, we offer a criterion for R to be L-semiprime. For a prime ring R, we characterizes Lie ideals L of R such that R is L-semiprime. Moreover, X-semiprimeness of matrix rings, prime rings (with a nontrivial idempotent), semiprime rings, regular rings, and subdirect products are studied.

Let D be a noncommutative division ring. In a recent paper, Lee and Lin proved that if charD≠2, the only solution of additive maps f,g on D satisfying the identity f(x)=xng(x−1) on D∖{0} with n≠2 a positive integer is the trivial case, that is, f=0 and g=0. Applying Hua's identity and the theory of functional and generalized polynomial identities, we give a complete solution of the same identity for any nonnegative integer n if charD=2.

We study the primeness of noncommutative polynomials on prime rings. Let [Formula: see text] be a prime ring with extended centroid [Formula: see text], [Formula: see text] a right ideal of [Formula: see text], [Formula: see text] a noncommutative polynomial over [Formula: see text], which is not a polynomial identity (PI) for [Formula: see text], and [Formula: see text]. Then [Formula: see text] for all [Formula: see text] if and only if one of the following holds: (i) [Formula: see text]; (ii) [Formula: see text] for some idempotent [Formula: see text] and [Formula: see text] such that either [Formula: see text] is a PI for [Formula: see text] or [Formula: see text] is central-valued on [Formula: see text] and [Formula: see text]. We then apply the result to higher commutators of right ideals. Some results of the paper are also studied from the view of point of the notion of [Formula: see text]-primeness of rings.

An element [Formula: see text] in a unital ring [Formula: see text] is said to have an inverse complement [Formula: see text] if [Formula: see text] is a unit of [Formula: see text] and [Formula: see text]. Unit-regular elements are studied from the viewpoint of the existence of inverse complements. As a source of unit-regular elements, we prove that if [Formula: see text] is a completely reducible submodule of [Formula: see text], then every element of [Formula: see text] is unit-regular if and only if any nonzero submodule of [Formula: see text] is not square zero. This generalizes some results due to Stopar in 2020. Finally, extending the case of real or complex matrices to the context of rings, we characterize the outer and reflexive inverses of a given unit-regular element depending only on its inverse complement.

Let [Formula: see text] be a simple algebra over its extended centroid [Formula: see text], and let [Formula: see text] be a noncommutative polynomial having zero constant term. We denote by [Formula: see text] the additive subgroup of [Formula: see text] generated by all elements [Formula: see text] for [Formula: see text]. It is proved that if [Formula: see text], then [Formula: see text] is equal to [Formula: see text], [Formula: see text], [Formula: see text], or [Formula: see text]. As to the case [Formula: see text], an example of a polynomial [Formula: see text] satisfying [Formula: see text] is given. Also, the polynomials [Formula: see text] with [Formula: see text] are characterized if [Formula: see text] and [Formula: see text]. Moreover, we work on the context of centrally closed prime algebras to get more general results.

We characterize bilinear functionals ϕ on a symmetric algebra A satisfying the two-sided zero product property (the 2-zpp, i.e., ϕ(x,y)=0 whenever xy=yx=0). If A is also a zero product determined algebra and if every derivation of the algebra A is inner, then A is a 2-zpd algebra (i.e., every bilinear functional on A satisfying the 2-zpp is of the form (x,y)↦τ1(xy)+τ2(yx) for x,y∈A, where τ1,τ2 are linear functionals on A). Conversely, if A is a finite-dimensional 2-zpd algebra, then the derivations of A are characterized, that is, given any derivation d of the algebra A, there exists a∈A such that, for all x∈A, d(x)−[a,x] lies in the Jacobson radical of A. Finally, we determine all bilinear functionals satisfying the 2-zpp on a specific zpd symmetric algebra and hence decide whether such an algebra is 2-zpd.

Let [Formula: see text] be an associative ring. Given a positive integer [Formula: see text], for [Formula: see text] we define [Formula: see text], the [Formula: see text]-generalized commutator of [Formula: see text]. By an [Formula: see text]-generalized Lie ideal of [Formula: see text] (at the [Formula: see text]th position with [Formula: see text]) we mean an additive subgroup [Formula: see text] of [Formula: see text] satisfying [Formula: see text] for all [Formula: see text] and all [Formula: see text], where [Formula: see text]. In the paper, we study [Formula: see text]-generalized commutators of rings and prove that if [Formula: see text] is a noncommutative prime ring and [Formula: see text], then every nonzero [Formula: see text]-generalized Lie ideal of [Formula: see text] contains a nonzero ideal. Therefore, if [Formula: see text] is a noncommutative simple ring, then [Formula: see text]. This extends a classical result due to Herstein [Generalized commutators in rings, Portugal. Math. 13 (1954) 137–139]. Some generalizations and related questions on [Formula: see text]-generalized commutators and their relationship with noncommutative polynomials are also discussed.

Let R be a ring having the property that every proper ideal of R is contained in a maximal ideal of R (in particular, if R is finitely generated as an ideal). Generalizing several known results, we characterize higher commutators V of R whenever R is generated by V (respectively, [R,V]) as an ideal. In particular, if V is a higher commutator of a unital ring R with 1∈V, then V is equal to either R or [R,R], or [[R,R],[R,R]]. Given a semiprime ring R, which is generated by [R,R], all higher commutators of R are obtained if R possesses a central higher commutator. We also characterize all higher commutators of Mn(D) for n≥2 when D is a unital commutative ring. In addition, if D is 2-torsion free and 2D⊊D, then M2(D) has infinitely many higher commutators.

For a subset [Formula: see text] of a ring [Formula: see text] we denote by [Formula: see text] the ideal of [Formula: see text] generated by [Formula: see text]. Given a higher commutator [Formula: see text] of [Formula: see text], if [Formula: see text] then [Formula: see text]? The question is motivated by the result that a ring [Formula: see text] is equal to its subring generated by [Formula: see text] if [Formula: see text] is either a noncommutative simple ring (by Herstein) or a unital ring with [Formula: see text] (by Eroǧlu). In this note, we study the question for the rings [Formula: see text] satisfying the property that every proper ideal of [Formula: see text] is contained in a maximal ideal (in particular, if [Formula: see text] is finitely generated as an ideal).

We give an example to show that, for nonunital rings [Formula: see text], the direct sum [Formula: see text] with [Formula: see text] regular has no in general right-left symmetry. It is then proved that the right-left symmetry actually holds in a left and right faithful ring.

Let R be an algebra. Given a noncommutative polynomial f, let f(R) stand for the additive subgroup of R generated by the image of f. For a unital or an affine algebra R, $$S_k(R)$$ is completely determined for any standard polynomial $$S_k$$ when R is generated by $$S_k(R)$$ as an ideal. Motivated by Brešar’s paper [Adv. Math. 374 (2020), 107346, 21 pp] and Robert’s paper [J. Oper. Theory 75 (2016), 387–408], under certain conditions we also prove that f(R) is equal to either [R, R] or the whole ring R. We obtain these results by studying the structure of Lie ideals L of a ring R whenever R is generated by [R, L] as an ideal.

AbstractLet R be a semiprime ring with extended centroid C and let $I(x)$ denote the set of all inner inverses of a regular element x in R. Given two regular elements $a, b$ in R, we characterise the existence of some $c\in R$ such that $I(a)+I(b)=I(c)$ . Precisely, if $a, b, a+b$ are regular elements of R and a and b are parallel summable with the parallel sum ${\cal P}(a, b)$ , then $I(a)+I(b)=I({\cal P}(a, b))$ . Conversely, if $I(a)+I(b)=I(c)$ for some $c\in R$ , then $\mathrm {E}[c]a(a+b)^{-}b$ is invariant for all $(a+b)^{-}\in I(a+b)$ , where $\mathrm {E}[c]$ is the smallest idempotent in C satisfying $c=\mathrm {E}[c]c$ . This extends earlier work of Mitra and Odell for matrix rings over a field and Hartwig for prime regular rings with unity and some recent results proved by Alahmadi et al. [‘Invariance and parallel sums’, Bull. Math. Sci.10(1) (2020), 2050001, 8 pages] concerning the parallel summability of unital prime rings and abelian regular rings.

Let [Formula: see text] be a semiprime ring, not necessarily with unity, with extended centroid [Formula: see text]. For [Formula: see text], let [Formula: see text] (respectively [Formula: see text], [Formula: see text]) denote the set of all outer (respectively inner, reflexive) inverses of [Formula: see text] in [Formula: see text]. In the paper, we study the inclusion properties of [Formula: see text], [Formula: see text] and [Formula: see text]. Among other results, we prove that for [Formula: see text] with [Formula: see text] von Neumann regular, [Formula: see text] (respectively [Formula: see text]) if and only if [Formula: see text] (respectively [Formula: see text]). Here, [Formula: see text] is the smallest idempotent in [Formula: see text] such that [Formula: see text]. This gives a common generalization of several known results.

Let R be a semiprime ring with maximal right ring of quo tients Q(mr) (R), and let n(1), n(2), ..., nk be k fixed positive integers. Sup pose that R is (n(1) +n(2) + ...+n(k))!-torsion free, and that f : rho -> Q(mr)(R) is an additive map, where rho is a nonzero right ideal of R. It is proved that if [[ ...[f(x), x(n1)], ...], x(n)k = 0 for all x is an element of rho, then [f (x), x] = 0 for all x is an element of p. This gives the result of Beidar et al. [2] for semiprime rings. Moreover, it is also proved that if R is p-torsion, where p is a prime integer with p = Sigma(k)(i=1) n(i), and if f -> : R -> Q(mr)(R) is an additive map satisfying [[ ...[f(x), x(n1)], ...], x(nk)] = 0 for all x is an element of R, then [f (x),x] = 0 for all x is an element of R.