We consider the following condition (*) on an associative ring R: (*). There exists a function f ... more We consider the following condition (*) on an associative ring R: (*). There exists a function f from R into R such that f is a group homomorphism of (R,+), f is injective on R 2,and f(xy) = (xy) n(x,y) for some positive integer n(x,y)> 1. Commutativity and structure are established for Artinian rings R satisfying (*), and a counterexample is given for non-Artinian rings. The results generalize commutativity theorems found elsewhere. The case n(x,y) = 2 is examined in detail. 2000 Mathematics Subject Classification: 16D70, 16P20. Let R be an associative ring, not necessarily with unity, and let R + denote the additive group of R. In[3], it was shown that R is commutative if it satisfies the following condition. (I) For each x and y in R, there exists n = n(x,y)> 1 such that (xy) n = xy. We generalize this result by considering the condition below. (II) There exists a function f from R into R such that f is a group homomorphism of R +, f is injective on R 2,andf(xy) = (xy) n(x,...
We show that universal algebras with factorable congruences such as rings with 1 and semirings wi... more We show that universal algebras with factorable congruences such as rings with 1 and semirings with 0 and 1 enjoy some of the properties of universal algebras whose congruence lattices are distributive, such as the strict refinement property and a variant of Jónsson's lemma.
Publisher Summary This chapter discusses coverings in the lattice of varieties. It discusses the ... more Publisher Summary This chapter discusses coverings in the lattice of varieties. It discusses the lattice of subvarieties of a given variety of universal algebras. A variety B immediately covers a variety A if A is a proper subvariety of B and if every member of B outside A generates B . The chapter discusses the connection between this covering relation and finite splitting algebras. The notion of immediate covers is utilized to show that for any given positive integer k , the variety of abelian groups of exponent 2 k is first order definable in the lattice of all group varieties. A critical algebra is a finite algebra that does not belong to the variety generated by its proper factors.
Transactions of the American Mathematical Society, 1971
A. The composition ringoid of functions on (i) the positive integers, (ii) all integers, (iii) th... more A. The composition ringoid of functions on (i) the positive integers, (ii) all integers, (iii) the reals and (iv) the complex numbers, do not satisfy any identities other than those satisfied by all composition ringoids. B. Given two words u, v of the free ringoid, specific functions on the positive integers, /!,...,A-can be described such that u(fu...,fk) and v(f,.. .,fk), evaluated at 1, are equal iff u = v is an identity of the free ringoid.
Transactions of the American Mathematical Society, 1974
The class of all rings that are Everett extensions of a ring in a variety U \mathfrak {U} by a ri... more The class of all rings that are Everett extensions of a ring in a variety U \mathfrak {U} by a ring in a variety B \mathfrak {B} is a variety U ⋅ B \mathfrak {U} \cdot \mathfrak {B} . With respect to this operation the set of all ring varieties is a partially ordered groupoid (under inclusion), that is not associative. A variety is idempotent iff it is the variety of all rings, or generated by a finite number of finite fields. No families of polynomial identities other than those equivalent to x = x x = x or x = y x = y are attainable on the class of all rings or on the class of all commutative rings.
Transactions of the American Mathematical Society, 1984
There are several generalizations to universal algebras of the notion "The group A \mathfrak... more There are several generalizations to universal algebras of the notion "The group A \mathfrak {A} is an extension of the group B \mathfrak {B} by the group C \mathfrak {C} ". In this paper we study three such generalizations and the corresponding products of classes of algebraic systems. Various results are presented. One such theorem characterizes the weakly congruence regular varieties admitting extensions of a particular sort. Another result gives, under a weak congruence permutability condition, an equational basis for the variety obtained by applying one such product to two other varieties.
B. Csákány. Ob abélévyh svojstvah primitivnyh klassov univérsal′nyh algébr (On the Abelian properties of the primitive classes of universal algebras). Acta scientiarum mathematicarum, vol. 25 (1964), pp. 202–208
A set A and an integer n > 1 are given. <S is any family of subsets of A n. Necessary and suffici... more A set A and an integer n > 1 are given. <S is any family of subsets of A n. Necessary and sufficient conditions are found for the existence of a set F of finitary partial operations on A such that S is the set of all subalgebras of <A; F} n. As a corollary, a family E of equivalence relations on A is the set of all congruences on <A; F} for some F if and only if E is an algebraic closure system on A 2 .
DEFINABILITY IN THE LATTICE OF RING VARIETIES AW AD A. ISKANDER A variety of associative rings B ... more DEFINABILITY IN THE LATTICE OF RING VARIETIES AW AD A. ISKANDER A variety of associative rings B immediately covers its subvariety A if every member of B outside A generates B. The variety {2x = 0, xy = 0} is the unique equationally complete variety with precisely two immediate covers in the lattice of all associative ring varieties. The variety of all Boolean rings is first order definable in the lattice of all associative ring varieties. So are the varieties defined by {2x = 0, xyyx = 0} and {xy = 0}.
Bijective functors σ and p are constructed between a category (R of commutative nonassociative ri... more Bijective functors σ and p are constructed between a category (R of commutative nonassociative rings and a category 9 of nilpotent groups, such that for all R e <R and G G S, P°R = R and σpG = G. Furthermore, if £ is a subclass of (R, then <£ has a decidable elementary theory iff σ<£ has a decidable elementary theory. Among the methods used to prove that certain classes of models have undecidable elementary theories are the "interpretation of one class into another" (cf. [7]), "semantic embedding" (cf. [1]), and "syntactic inclusion" (cf. [3]). MaFcev [3] used "syntactic inclusion" to show that the class of all nilpotent class Az-groups (for n>2) has an undecidable elementary theory. In fact, MaPcev employed a notion which we will here call "syntactic isomorphism". If two classes of models are syntactically isomorphic, then their elementary theories are either simultaneously undecidable or simultaneously decidable. One of the results of [3] states that the ring of integers Z is syntactically isomorphic to the free nilpotent class 2-group on two free generators, and the class of all not necessarily associative rings with identity is syntactically isomorphic to a class of nilpotent class 2-groups with two constants. In the present paper, we show that certain classes of rings with identity are syntactically isomorphic to classes of nilpotent groups whose nilpotency class is at most 3; in particular, the ring of integers Z is syntactically isomorphic to the free nilpotent class 3-group with two free generators, and the class of all Boolean algebras is syntactically isomorphic to a class of nilpotent class 2-groups. The phrase "nonassociative ring" will mean "a ring with identity (denoted by 1) that is not necessarily associative". The notations of decidability theory that will be used here are those of [7]. Unless otherwise stated, the notations of group theory that are used here are those of [4]. / Let JC be a class of models of type r. The first-order language of type τ will be denoted by Lr. The elementary theory of JC, i.e., all sentences of Lr that
Every nonassociative Boolean domain can be embedded into a Boolean quasitield. A similar result h... more Every nonassociative Boolean domain can be embedded into a Boolean quasitield. A similar result holds for medial nonassociative Boolean domains. We describe connections between medial nonassociative Boolean rings and associative rings in which squaring is an automorphism. The lattice of varieties of such associative rings and the lattice of varieties of medial nonassociative Boolean rings are isomorphic to the one point completion of the lattice of finite factor closed sets of positive integers. We describe the free members of such varieties.
We consider the following condition (*) on an associative ring R: (*). There exists a function f ... more We consider the following condition (*) on an associative ring R: (*). There exists a function f from R into R such that f is a group homomorphism of (R,+), f is injective on R 2,and f(xy) = (xy) n(x,y) for some positive integer n(x,y)> 1. Commutativity and structure are established for Artinian rings R satisfying (*), and a counterexample is given for non-Artinian rings. The results generalize commutativity theorems found elsewhere. The case n(x,y) = 2 is examined in detail. 2000 Mathematics Subject Classification: 16D70, 16P20. Let R be an associative ring, not necessarily with unity, and let R + denote the additive group of R. In[3], it was shown that R is commutative if it satisfies the following condition. (I) For each x and y in R, there exists n = n(x,y)> 1 such that (xy) n = xy. We generalize this result by considering the condition below. (II) There exists a function f from R into R such that f is a group homomorphism of R +, f is injective on R 2,andf(xy) = (xy) n(x,...
We show that universal algebras with factorable congruences such as rings with 1 and semirings wi... more We show that universal algebras with factorable congruences such as rings with 1 and semirings with 0 and 1 enjoy some of the properties of universal algebras whose congruence lattices are distributive, such as the strict refinement property and a variant of Jónsson's lemma.
Publisher Summary This chapter discusses coverings in the lattice of varieties. It discusses the ... more Publisher Summary This chapter discusses coverings in the lattice of varieties. It discusses the lattice of subvarieties of a given variety of universal algebras. A variety B immediately covers a variety A if A is a proper subvariety of B and if every member of B outside A generates B . The chapter discusses the connection between this covering relation and finite splitting algebras. The notion of immediate covers is utilized to show that for any given positive integer k , the variety of abelian groups of exponent 2 k is first order definable in the lattice of all group varieties. A critical algebra is a finite algebra that does not belong to the variety generated by its proper factors.
Transactions of the American Mathematical Society, 1971
A. The composition ringoid of functions on (i) the positive integers, (ii) all integers, (iii) th... more A. The composition ringoid of functions on (i) the positive integers, (ii) all integers, (iii) the reals and (iv) the complex numbers, do not satisfy any identities other than those satisfied by all composition ringoids. B. Given two words u, v of the free ringoid, specific functions on the positive integers, /!,...,A-can be described such that u(fu...,fk) and v(f,.. .,fk), evaluated at 1, are equal iff u = v is an identity of the free ringoid.
Transactions of the American Mathematical Society, 1974
The class of all rings that are Everett extensions of a ring in a variety U \mathfrak {U} by a ri... more The class of all rings that are Everett extensions of a ring in a variety U \mathfrak {U} by a ring in a variety B \mathfrak {B} is a variety U ⋅ B \mathfrak {U} \cdot \mathfrak {B} . With respect to this operation the set of all ring varieties is a partially ordered groupoid (under inclusion), that is not associative. A variety is idempotent iff it is the variety of all rings, or generated by a finite number of finite fields. No families of polynomial identities other than those equivalent to x = x x = x or x = y x = y are attainable on the class of all rings or on the class of all commutative rings.
Transactions of the American Mathematical Society, 1984
There are several generalizations to universal algebras of the notion "The group A \mathfrak... more There are several generalizations to universal algebras of the notion "The group A \mathfrak {A} is an extension of the group B \mathfrak {B} by the group C \mathfrak {C} ". In this paper we study three such generalizations and the corresponding products of classes of algebraic systems. Various results are presented. One such theorem characterizes the weakly congruence regular varieties admitting extensions of a particular sort. Another result gives, under a weak congruence permutability condition, an equational basis for the variety obtained by applying one such product to two other varieties.
B. Csákány. Ob abélévyh svojstvah primitivnyh klassov univérsal′nyh algébr (On the Abelian properties of the primitive classes of universal algebras). Acta scientiarum mathematicarum, vol. 25 (1964), pp. 202–208
A set A and an integer n > 1 are given. <S is any family of subsets of A n. Necessary and suffici... more A set A and an integer n > 1 are given. <S is any family of subsets of A n. Necessary and sufficient conditions are found for the existence of a set F of finitary partial operations on A such that S is the set of all subalgebras of <A; F} n. As a corollary, a family E of equivalence relations on A is the set of all congruences on <A; F} for some F if and only if E is an algebraic closure system on A 2 .
DEFINABILITY IN THE LATTICE OF RING VARIETIES AW AD A. ISKANDER A variety of associative rings B ... more DEFINABILITY IN THE LATTICE OF RING VARIETIES AW AD A. ISKANDER A variety of associative rings B immediately covers its subvariety A if every member of B outside A generates B. The variety {2x = 0, xy = 0} is the unique equationally complete variety with precisely two immediate covers in the lattice of all associative ring varieties. The variety of all Boolean rings is first order definable in the lattice of all associative ring varieties. So are the varieties defined by {2x = 0, xyyx = 0} and {xy = 0}.
Bijective functors σ and p are constructed between a category (R of commutative nonassociative ri... more Bijective functors σ and p are constructed between a category (R of commutative nonassociative rings and a category 9 of nilpotent groups, such that for all R e <R and G G S, P°R = R and σpG = G. Furthermore, if £ is a subclass of (R, then <£ has a decidable elementary theory iff σ<£ has a decidable elementary theory. Among the methods used to prove that certain classes of models have undecidable elementary theories are the "interpretation of one class into another" (cf. [7]), "semantic embedding" (cf. [1]), and "syntactic inclusion" (cf. [3]). MaFcev [3] used "syntactic inclusion" to show that the class of all nilpotent class Az-groups (for n>2) has an undecidable elementary theory. In fact, MaPcev employed a notion which we will here call "syntactic isomorphism". If two classes of models are syntactically isomorphic, then their elementary theories are either simultaneously undecidable or simultaneously decidable. One of the results of [3] states that the ring of integers Z is syntactically isomorphic to the free nilpotent class 2-group on two free generators, and the class of all not necessarily associative rings with identity is syntactically isomorphic to a class of nilpotent class 2-groups with two constants. In the present paper, we show that certain classes of rings with identity are syntactically isomorphic to classes of nilpotent groups whose nilpotency class is at most 3; in particular, the ring of integers Z is syntactically isomorphic to the free nilpotent class 3-group with two free generators, and the class of all Boolean algebras is syntactically isomorphic to a class of nilpotent class 2-groups. The phrase "nonassociative ring" will mean "a ring with identity (denoted by 1) that is not necessarily associative". The notations of decidability theory that will be used here are those of [7]. Unless otherwise stated, the notations of group theory that are used here are those of [4]. / Let JC be a class of models of type r. The first-order language of type τ will be denoted by Lr. The elementary theory of JC, i.e., all sentences of Lr that
Every nonassociative Boolean domain can be embedded into a Boolean quasitield. A similar result h... more Every nonassociative Boolean domain can be embedded into a Boolean quasitield. A similar result holds for medial nonassociative Boolean domains. We describe connections between medial nonassociative Boolean rings and associative rings in which squaring is an automorphism. The lattice of varieties of such associative rings and the lattice of varieties of medial nonassociative Boolean rings are isomorphic to the one point completion of the lattice of finite factor closed sets of positive integers. We describe the free members of such varieties.
Uploads
Papers by Awad Iskander