Rings with Boolean Lattices of One-Sided Annihilators

The present paper is part of the research on the description of rings with a given property of the lattice of left (right) annihilators. The anti-isomorphism of lattices of left and right annihilators in any ring gives some kind of symmetry: the lattice of left annihilators is Boolean (complemented, distributive) if and only if the lattice of right annihilators is such. This allows us to restrict our investigations mainly to the left side. For a unital associative ring R, we prove that the lattice of left annihilators in R is Boolean if and only if R is a reduced ring. We also prove that the lattice of left annihilators of R being two-sided ideals is complemented if and only if this lattice is Boolean. The last statement, in turn, is known to be equivalent to the semiprimeness of R. On the other hand, for any complete lattice L, we construct a nilpotent ring whose lattice of left annihilators coincides with its sublattice of left annihilators being two-sided ideals and is isomorphic to L. This construction shows that the assumption of R being unital cannot be dropped in any of the above two results. Some additional results on rings with distributive or complemented lattices of left annihilators are obtained.

Differential polynomial rings in several variables over locally nilpotent rings

We show that a differential polynomial ring over a locally nilpotent ring in several commuting variables is Behrens radical, extending a result by Chebotar.

Notes on topological rings

We prove that every infinite nilpotent ring R admits a ring topology T for which (R, T ) has an open totally bounded countable subring with trivial multiplication. A new example of a compact ring R for which R2 is not closed, is given. We prove that every compact Bezout domain is a principal ideal domain.

The Identities of Additive Binary Arithmetics

Operations of arbitrary arity expressible via addition modulo $2^n$ and bitwise addition modulo $2$ admit a simple description. The identities connecting these two additions have a finite basis. Moreover, the universal algebra $\mathbb{Z}/2^n\mathbb{Z}$ with these two operations is rationally equivalent to a nilpotent ring and, therefore, generates a Specht variety.

A Kind of Graph Structure on Non-reduced Rings

In this paper, a kind of graph structure ΓN(R) of a ring R is introduced, and the interplay between the ring-theoretic properties of R and the graph-theoretic properties of ΓN(R) is investigated. It is shown that if R is Artinian or commutative, then ΓN(R) is connected, the diameter of ΓN(R) is at most 3; and if ΓN(R) contains a cycle, then the girth of ΓN(R) is not more than 4; moreover, if R is non-reduced, then the girth of ΓN(R) is 3. For a finite commutative ring R, it is proved that the edge chromatic number of ΓN(R) is equal to the maximum degree of ΓN(R) unless R is a nilpotent ring with even order. It is also shown that, with two exceptions, if R is a finite reduced commutative ring and S is a commutative ring which is not an integral domain and ΓN(R) ≃ ΓN(S), then R ≃ S. If R and S are finite non-reduced commutative rings and ΓN(R) ≃ ΓN(S), then |R|=|S| and |N(R)|=|N(S)|.

On orders of directly indecomposable finite rings

Let R be a directly indecomposable finite ring. Let p be a prime, let m be a positive integer and suppose the radical of R has pm elements. Then we show that . As a consequence, we have that, for a given finite nilpotent ring N, there are up to isomorphism only finitely many finite rings not having simple ring direct summands, with radical isomorphic to N. Let R* denote the group of units of R. Then we prove that (1 − 1/p)m+1 ≤ |R*| / |R| ≤ 1 − 1/pm. As a corollary, we obtain that if R is a directly indecomposable non-simple finite 2′-ring then |R| < |R*| |Rad(R)|.

Groups Admitting only Finitely Many Nilpotent Ring Structures

The abelian groups which are the additive groups of only finitely many non-isomorphic (associative) nilpotent rings are studied. Progress is made toward a complete classification of these groups. In the torsion free case, the actual number of non-isomorphic nilpotent rings these groups support is obtained.

A New Bound for Nil U-Rings

A U-ring is a ring in which every subring is a meta ideal. A meta ideal of a ring R is a subring I of R which lies in a chain of subrings,with the properties:(1) Iλ is an ideal of Iλ+1 for all λ < β;(2) If α is a limit ordinal number, then Iα = ∪λ<αIλ.Freidman [3] proved that every nil U-ring is a locally nilpotent ring. Since there are many locally nilpotent rings which are not U-rings, the class of locally nilpotent rings is not a very good bound for the class of nil U-rings. This paper establishes a new bound for nil U-rings based on a property of the multiplicative semigroup of the ring.