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Nearly $60$ years ago, L\'{a}szl\'{o} Fuchs posed the problem of determining which groups can be realized as the group of units of a commutative ring. To date, the question remains open, although significant progress has been made. Along this line, one could also ask the more general question as to which finite groups can be realized as the group of units of a finite ring. In this paper, we consider the question of which $2$-groups are realizable as unit groups of finite rings, a necessary step toward determining which nilpotent groups are realizable. We prove that all $2$-groups of exponent $4$ and exponent $2$ are realizable in characteristic $2$, and we prove that many $2$-groups with exponent $4$ and nilpotency class $3$ are realizable in characteristic $2$. On the other hand, we provide an example of a $2$-group with exponent $4$ and nilpotency class $4$ that is not realizable in characteristic $2$. Moreover, while some groups of exponent greater than $4$ are realizable as unit groups of rings, we prove that any $2$-group with a self-centralizing element of order $8$ or greater is never realizable in characteristic $2^m$, and consequently any indecomposable, nonabelian group with a self-centralizing element of order $8$ or greater cannot be the group of units of a finite ring., Comment: 27 pages. The main theorem in the previous version was incorrect; specifically, Proposition 3.9 was incorrect. Pages 9-13 of this version include new content dedicated to the proof of Theorem 1.2 (the updated main result), and Section 4 is dedicated to a counterexample to the main theorem in the previous version. Question 6.8 is also new

We determine explicitly the Gauss sum τ l ( χ , e ) = ∑ X ∈ GL 2 ( Z / p l Z ) χ ( X ) e ( Tr X ) on the general linear group GL 2 ( Z / p l Z ) for every irreducible character χ of GL 2 ( Z / p l Z ) and a nontrivial additive character e of Z / p l Z , where p is an odd prime and l is an integer ⩾2. While there are several studies of the Gauss sums on finite algebraic groups defined over a finite field, this paper seems to be the first one which determines the Gauss sums on a matrix group over a finite ring.

It is shown that, for a fixed positive integer g, there are finitely many isomorphism classes of rings whose zero-divisor graph has genus g. The proof can then be modified to yield an analogous result for nonorientable genus.

The zero-divisor graph of a ring R is defined as the directed graph Γ ( R ) that its vertices are all non-zero zero-divisors of R in which for any two distinct vertices x and y, x → y is an edge if and only if x y = 0 . Recently, it has been shown that for any finite ring R, Γ ( R ) has an even number of edges. Here we give a simple proof for this result. In this paper we investigate some properties of zero-divisor graphs of matrix rings and group rings. Among other results, we prove that for any two finite commutative rings R , S with identity and n , m ⩾ 2 , if Γ ( M n ( R ) ) ≃ Γ ( M m ( S ) ) , then n = m , | R | = | S | , and Γ ( R ) ≃ Γ ( S ) .

Throughout this paper the symbol R will be used to denote an associative ring with a multiplicative identity. R,[R”] denotes the subring of R which is generated over R, by R”, where R, is the prime subring of R and R” is the group of units of R. We use J(R) to denote the Jacobson radical of R. In Section 1 we prove that if R is a finite ring and at most one simple component of the semi-simple ring R/J(R) is a field of order 2, then R” is a nilpotent group iff R is a direct sum of two-sided ideals that are homomorphic images of group algebras of type SP, where S is a particular commutative finite ring, P is a finite p-group, and p is a prime number. In Section 2 we study the artinian rings R (i.e., rings with minimum condition) with RO a finite ring, R finitely generated over its center and so that every block of R is as in Lemma 1.1. We note that every finite-dimensional algebra over a field F of characteristic p > 0, so that F is not the finite field of order 2 is of this type. We prove that R is of this type and R” is a nilpotent group iff R is a direct sum of two-sided ideals that are particular homomorphic images of crossed products. Finally we present an analogous theorem when R/J(R) is a commutative ring. The author acknowledges that various results of this paper were obtained under the supervision of Professor N. Popescu.

(2) R is a finite ring. Indeed (see, for instance, [5]), let I R j = r < 03. Consider fi : R -+ R, i = I,2 ,..., given by fi(a) = ui for all a E R. We have no more than F-I distinct maps fi . So fit = fi for some I < K. Thus ux= a2 for all a E R and we take f(x) = .G’ 9. We remark that varieties of the form var(R), where R is a finite ring, were described in many ways among all other varieties of rings by Lvov [5, 61 and Kruse [7]. Our main result is the following

Let R be a finite ring with identity, T its set of idempotents. We study the subsets of T that can be closed under multiplication and the implications that fact has to the structure of R. We consider the subset M of all minimal idempotents and zero and prove that M is closed under multiplication if and only if R is a direct sum of local rings. We achieve this by studying the properties of units that are preserved by idempotents.

I prove that given a finite semigroup or finite associative ring S and a system Σ of equations of the form ax = b or xa = b , where a , b ∈ S , x is an unknown, it is algorithmically impossible to decide whether or not Σ is solvable over S , that is, whether or not there exists a bigger semigroup or ring (resp. finite semigroup, finite ring) T > S such that Σ has a solution in T . The proof employs the unsolvability of the uniform word problem in the case of groups (Novikov) and in the class of finite groups (Slobodskoii) and the so-called split systems.

If R is a homomorphic image of a finite Frobenius local ring, there is a known construction that produces Latin square type partial difference sets (PDS) in R×R. By a simple construction, we show that every finite ring is a homomorphic image of a finite Frobenius ring and every finite local ring is a homomorphic image of a finite Frobenius local ring. Consequently, Latin square type PDS can be constructed in R×R for any finite local ring R, where the additive group (R,+) can be any finite abelian p-group.