Semiprimality and Nilpotency of Nonassociative Rings Satisfying x(yz) = y(zx)

In this article we study nonassociative rings satisfying the polynomial identity x(yz) = y(zx), which we call “cyclic rings.” We prove that every semiprime cyclic ring is associative and commutative and that every cyclic right-nilring is solvable. Moreover, we find sufficient conditions for the nilpotency of cyclic right-nilrings and apply these results to obtain sufficient conditions for the nilpotency of cyclic right-nilalgebras.     

Groupoids with the identity defining the commutative Moufang loops

It is known that in the class of loops the identity xy · xz = x ² · yz defines the commutative Moufang loops. In the article, we consider groupoids with this identity and establish properties of some classes of such groupoids.     

Varieties of idempotent slim groupoids

Idempotent slim groupoids are groupoids satisfying xx ≈ x and x(yz) ≈ xz. We prove that the variety of idempotent slim groupoids has uncountably many subvarieties. We find a four-element, inherently nonfinitely based idempotent slim groupoid; the variety generated by this groupoid has only finitely many subvarieties. We investigate free objects in some varieties of idempotent slim groupoids determined by permutational equations. The work is a part of the research project MSM0021620839 financed by MSMT and partly supported by the Grant Agency of the Czech Republic, grant #201/05/0002.     

On Maps Determined by Zero Products

The main result describes a bijective additive map h between prime rings with nontrivial idempotents that satisfies h(x)h(y)h(z) = 0 whenever xy = yz = 0. The proof is based on the consideration of a multiadditive map satisfying a related condition.     

Slim groupoids

Slim groupoids are groupoids satisfying x(yz) ≈ xz. We find all simple slim groupoids and all minimal varieties of slim groupoids. Every slim groupoid can be embedded into a subdirectly irreducible slim groupoid. The variety of slim groupoids has the finite embeddability property, so that the word problem is solvable. We introduce the notion of a strongly nonfinitely based slim groupoid (such groupoids are inherently nonfinitely based) and find all strongly nonfinitely based slim groupoids with at most four elements; up to isomorphism, there are just two such groupoids. The work is a part of the research project MSM0021620839 financed by MSMT.     

On hyperassociativity

The paper shows that the identitiesx ⁴ x ²,xyxzxyx xyzyx, xy ² z ² xyz ² yz ² andx ₂ y ² z x ₂ yx ₂ yz form an equational basis of H, the largest hyperassociative variety of semigroups. We present here a model of the free algebra in H on 2 generators (it has 94 elements) and solve the word problem for completely regular part of the free algebra in H with any number of generators.Presented by B. M. Schein.The author acknowledges the support of the Grant no. 201/93/2121 of the Grant Agency of Czech Republic. The main part of research was carried out during the author's visit to Prof. I. G. Rosenberg at the University of Montreal within the NATO Collaborative Grant no. LG 930 302.     

Square-zero derivations of matrix algebras over commutative rings

Let R be a commutative ring with identity, M _n (R) the R-algebra consisting of all n by n matrices over R. In this article, for n ≥ 5 we classify linear maps φ from M _n (R) into itself satisfying φ(x)x + xφ(x) = 0 whenever x ² = 0. We call such maps as square-zero derivations.     

Existence of the non-radially symmetric ground state for p-Laplacian equations involving Choquard type

We consider some p-Laplacian type equations with sum of nonlocal term and subcritical nonlinearities. We prove the existence of the ground states, which are positive. Because of including p=2, these results extend the results of Li, Ma and Zhang [Nonlinear Analysis: Real World Application 45(2019) 1-25]. When p=2, N=3, by a variant variational identity and a constraint set, we can prove the existence of a non-radially symmetric solution. Moreover, this solution u(x₁, x₂, x₃) is radially symmetric with respect to (x₁, x₂) and odd with respect to x₃.     

On directed zero-divisor graphs of finite rings

For an artinian ring R, the directed zero-divisor graph Γ(R) is connected if and only if there is no proper one-sided identity element in R. Sinks and sources are characterized and clarified for a finite ring R. Especially, it is proved that for any ring R, if there exists a source y in Γ(R) with y²=0, then |R|=4 and R={0,x,y,z}, where x and z are left identity elements and yx=0=yz. Such a ring R is also the only ring such that Γ(R) has exactly one source. This shows that Γ(R) cannot be a network for any finite or infinite ring R.     

An elegant 3-basis for inverse semigroups

It is well known that in every inverse semigroup the binary operation and the unary operation of inversion satisfy the following three identities:

Algebres de rang 3 de composition faible

We study algebras of rank three admitting a nonzero quadratic form ?which satisfies the identity ?(x ²) = ?(x ²), and we show the relationship with power-associativity. By means of the Wedderburn decomposition relative to their radical, we prove that power-associative rank three algebras are actually Jordan.     

The varieties of loops of Bol-Moufang type

A loop identity is of Bol-Moufang type if two of its three variables occur once on each side, the third variable occurs twice on each side, and the order in which the variables appear on both sides is the same, viz. ((xy)x)z = x(y(xz)). Loop varieties defined by one identity of Bol-Moufang type include groups, Bol loops, Moufang loops and C-loops. We show that there are exactly 14 such varieties, and determine all inclusions between them, providing all necessary counterexamples, too. This extends and completes the programme of Fenyves [Fe69]. Received October 23, 2003; accepted in final form April 12, 2005.     

Solvability of a Commutative Algebra which Satisfies (x 2)2 = 0

We studied the solvability of the algebra which satisfies the polynomial identity (x ²)² = 0. We believe that, if A is a finite dimensional commutative algebra over a field F of characteristic not 2 which satisfies (x ²)² = 0 for all x ∈ A, then A is solvable. In this article we proved this when dim  _F A ≤ 7.     

Characterizations of generalized quadrangles by generalized homologies

Let S = (P, B, I) be a generalized quadrangle of order (s, t). For x, y P, x y, let (x, y) be the group of all collineations of S fixing x and y linewise. If z {x, y}, then the set of all points incident with the line xz (resp. yz) is denoted by (resp. ). The generalized quadrangle S = (P, B, I) is said to be (x, y)-transitive, x y, if (x, y) is transitive on each set and . If S = (P, B, I) is a generalized quadrangle of order (s, t), s > 1 and t > 1, which is (x, y)-transitive for all x, y P with x y, then it is proved that we have one of the following: (i) S W(s), (ii) S Q(4, s), (iii) S H(4, s), (iv) S Q(5, s), (v) s = t² and all points are regular.     

Étude des ω-PI Algèbres Commutatives de Degré 4: II. Algèbres Non Barycentriques Invariantes par Gamétisation

In this article we study the algebras satisfying the ω-polynomial identity x ² x ² ? x ⁴ = δ(x ² ? x) with δ ≠ 0 but do not satisfy any monomial identities of degree ≤4. We show that there exist such algebras for all δ ≠ 0 and they have a unique baric function. We give conditions for the existence of idempotents of weight 0 or 1, and we construct the three Peirce decompositions associated to these idempotent elements.     

Generalized derivations and left multipliers on Lie ideals

Let m, n be two fixed positive integers and let R be a 2-torsion free prime ring, with Utumi quotient ring U and extended centroid C. We study the identity F(x ^m+n+1) = F(x)x ^m+n  + x ^m D(x)x ⁿ for x in a non-central Lie ideal of R, where both F and D are generalized derivations of R and then determine the relationship between the form of F and that of D. In particular the conclusions of the main theorem say that if D is the non-zero map in R, then R satisfies the standard identity s ₄(x ₁, . . . , x ₄) and D is a usual derivation of R.     

Admissible Orders of Jordan Loops

A commutative loop is Jordan if it satisfies the identity x²(yx) = (x²y)x. Using an amalgam construction and its generalizations, we prove that a nonassociative Jordan loop of order n exists if and only if n≧ 6 and n≠ 9. We also consider whether powers of elements in Jordan loops are well‐defined, and we construct an infinite family of finite simple nonassociative Jordan loops. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 103–118, 2009     

Comparison semigroups and algebras of transformations

We characterize algebras of transformations on a set under the operations of composition and the pointwise switching function defined as follows: (f,g)[h,k](x)=h(x) if f(x)=g(x), and k(x) otherwise. The resulting algebras are both semigroups and comparison algebras in the sense of Kennison. The same characterization holds for partial transformations under composition and a suitable generalisation of the quaternary operation in which agreement of f,g includes cases where neither is defined. When a zero element is added (modelling the empty function), the resulting signature is rich enough to encompass many operations on semigroups of partial transformations previously considered, including set difference and intersection, restrictive product, and a functional analog of union. When an identity element is also added (modelling the identity function), further domain-related operations can be captured.     

Some Examples of Simple Small Singularities

We provide some examples of simple small singularities of higher dimensional algebraic varieties. One of them is an E ₆ type singularity w ² ? z ³ + xy ³ ? 3x ² yz ? x ⁵ + xzw ? x ⁴ y = 0 in ?⁴. We also treat small contractions of curves with higher genera whose normal bundles are not negative.