A NOTE ON GROUP RINGS WITH TRIVIAL UNITS

In 1963, Gilmer characterized all finite commutative rings with a cyclic group of units and, in 1967, Eldridge and Fischer generalized these results to rings with minimum condition. In the present paper these results are extended to semiperf ect rings and generalizations of the three theorems are obtained. It is shown that a semiperfect ring with cyclic group of units is finite and is either commutative or is the direct sum of a commutative ring and the 2 x 2 upper triangular matrix ring over the field of two elements. Let R be semiperfect with an abelian group of units. It is shown that R is finite if either the group of units is finite or the group of units is finitely generated and the Jacobson radical is nil. The proofs of all these results depend on our main theorem: The structure of a semiperfect ring R with an abelian group of units is described completely up to the structure of commutative local rings. (That is commutative rings with a unique maximal ideal.) The groups of units of these local rings are shown to be direct factors of the group of units of R.

In this study, we construct a new public key cryptosystem both like El-Gamal and RSA cryptosystem based on Bass cyclic units in integral group rings. Naturally, the underlying hard problems for this system are both decisional Diffie-Hellman assumption and factorization of integers. Since the system is related to RSA scheme indirectly, key generation process is very efficient. This cryptosystem actually supports and strengthens the current public key cryptosystems in [6], [8] and [20]. Encryption algorithm in the system is constructed via non-commutative operations in integral group rings of dihedral groups. Finally, we briefly discuss the security of our cryptosystem against some known attacks.

In this paper some conditions for a skew group ring or a crossed product to have finite weak global dimension are given.Using these results we obtain some necessary conditions and some sufficient conditions for a skew group ring or a crossed product to be a Dubrovin valuation ring.If R*G is a skew group ring, where the coefficient ring R is a commutative ring and G is a finite group, then we prove that the conditions we obtained become necessary and sufficient conditions.In particular, if R is a commutative valuation ring, then R*G is a Dubrovin valuation ring if and only if G T=<1>,where G T is the inertial group of R.

Two questions on semi-clean group rings

We calculate, only in terms of a commutative unital ring R of prime characteristic p and an abelian p-mixed group G, the classical Warfield q-invariants Wα,q(VR(G)) of the group VR(G) of all normalized units in the group ring R(G). This continues our results in (Extr. Math., 2005), (Collect. Math., 2008) and (J. Alg. Appl., 2008).

In this note, we show that when G G is a torsion group the second center of the group of units U ( Z G ) U({\mathbb Z}G) of the integral group ring Z G {\mathbb Z}G is generated by its torsion subgroup and by the center of U ( Z G ) U({\mathbb Z}G) . This extends a result of Arora and Passi (1993) from finite groups to torsion groups, and completes the characterization of hypercentral units in Z G {\mathbb Z}G when G G is a torsion group.

A commutative ring R is called J-rad clean in case, for any r ∈ R, there is an idempotent e ∈ R such that r−e ∈ U(R) and re ∈ J(R), where U(R) and J(R) denote the set of units and the Jacobson radical of R, respectively. Also, a ring R is called semiregular if R/J(R) is regular in the sense of von Neumann and idempotents lift modulo J(R). We demonstrate that these two concepts are, actually, equivalent and we portray a portion of the properties of this class of rings. In particular, as a direct application, we prove that the commutative group ring RG is J-rad clean if, and only if, R is a commutative J-rad clean ring and G is a torsion abelian group, provided that J(R) is nil.

Let $KG$ be the group ring of a group $G$ over a commutative ring $K$ with unity. The rings $KG$ are described for which $xx^\sigma =x^\sigma x$ for all $x=\sum _{g\in G}\alpha _gg\in KG$, where $x\mapsto x^\sigma =~\sum _{g\in G}\alpha _gf(g)\sigma (g)$ is an involution of $KG$; here $f: G\to U(K)$ is a homomorphism and $\sigma$ is an antiautomorphism of order two of $G$.

Let B be any nite commutative ring with identity. In this case, B(X; 1 p kZ0) B(X; 1 p 2Z0) B(X; 1 p Z0), where p is a prime number and k 1, is the descending chain of commutative semigroup rings. All these semigroup rings are containing the polynomial ring B(X;Z0). In this paper, we introduced a construction technique of cyclic codes through the semi- group ring B(X; 1 p kZ0) instead of a polynomial ring.

Units in regular abelian p-group rings

It is known that if the unit group of an integral group ring ZG is trivial, then the unit group of Z(G × C 2 ) is trivial as well [3] . The aim of this study is twofold: firstly, to identify rings R that are D-adapted for the direct product D = G × H of abelian groups G and H, such that the unit group of the ring R(G × H) is trivial. Our second objective is to investigate the necessary and sufficient conditions on both the ring R and the direct factors of D to satisfy the property that the normalized unit group V (RD) is trivial in the case where D is one of the groups G × C 3 , G × K 4 or G × C 4 , where G is an arbitrary finite abelian group, C n denotes a cyclic group of order n and K 4 is Klein 4-group. Hence, the study extends the related result in [18] . The author wishes to express gratitude to the referees for their valuable suggestions and to all the members of the Algebra and Discrete Mathematics community for their support and contributions.

<span>The notion of the matrix Jacobson graph was introduced in 2019. Let </span><span class="math inline"><em>R</em></span><span> be a commutative ring and </span><span class="math inline"><em>J</em>(<em>R</em>)</span><span> be the Jacobson radical of ring </span><span class="math inline"><em>R</em></span><span>. The matrix Jacobson graph of ring </span><span class="math inline"><em>R</em></span><span> size </span><span class="math inline"><em>m</em> × <em>n</em></span><span>, denoted </span><span class="math inline">𝔍(<em>R</em>)^{<em>m</em> × <em>n</em>}</span><span>, is defined as a graph where the vertex set is </span><span class="math inline"><em>R</em>^{<em>m</em> × <em>n</em>} ∖ <em>J</em>(<em>R</em>)^{<em>m</em> × <em>n</em>}</span><span> such that two distinct vertices </span><span class="math inline"><em>A</em>, <em>B</em></span><span> are adjacent if and only if </span><span class="math inline">1 − det(<em>A</em>^<em>t</em><em>B</em>)</span><span> is not a unit in ring </span><span class="math inline"><em>R</em></span><span>. Here we obtain some graph theoretical properties of </span><span class="math inline">𝔍(<em>R</em>)^{<em>m</em> × <em>n</em>}</span><span> including its connectivity, planarity and perfectness.</span>

The Grothendieck ring of a finite group

We fully describe all normal complements of the trivial units in the integral group rings of the dihedral groups of order 6 and 8.

Let S be a cancellative monoid with quotient group of torsion-free rank a.We show that the monoid ring R[S] is a Hilbert ring if and only if the polynomial ring R[{ X, },s/] is a Hilbert ring, where |/| = a.Assume that R is a commutative unitary ring and G is an abelian group.The first research problem listed in [K, Chapter 7] is that of determining equivalent conditions in order that the group ring R[G] should be a Hilbert ring.Matsuda has considered this question in [Ml, 6].His results show that for G finitely generated, R[G] and R are simultaneously Hilbert rings; if G is not finitely generated, then R[G] a Hilbert ring implies that R is a Hilbert ring, but the converse fails.In [M2, 5] Matsuda considers briefly the corresponding Hilbert-ring-characterization problem for a monoid ring R[S], where S is torsion-free and cancellative.Let a be the torsion-free rank of G and let {Xj)lsl be a set of indeterminates over R of cardinality a.In Corollary 1 we show that R[G] and P[{^,},e/] are simultaneously Hilbert rings.While the problem of determining equivalent conditions for P[{Ar,}/e/] to be a Hilbert ring has not been completely resolved, it has been worked on extensively [Kr, Go, L, Gl, H], and a significant body of positive results exists concerning this problem.The proof of Theorem 1 suggests the following conjecture: if S is a cancellative commutative monoid with quotient group G, then R[S] is a Hilbert ring if and only if R[G] is a Hilbert ring; this result is established in Theorem 2.All monoids considered are assumed to be commutative, and rings are assumed to be commutative and unitary.The statement of Theorem 1 uses the following terminology.For a cardinal number a, an extension ring T of R is said to be a-generated over R if T is generated over R by a set of cardinality at most a.Theorem 1. Assume that X = {XA is a set of indeterminates of cardinality a over the ring R. Denote by Za the direct sum of a copies of the additive group Z of integers.The following conditions are equivalent.(1) R[X] is not a Hilbert ring.(2) There exists a prime ideal P of R such that R/P admits an a-generated extension ring that is a G-domain, but not afield.(3) The group ring R[Za] is not a Hilbert ring._