The Number of Subgroups of PSL(2, Z) When Acting on F p ∪ {∞}

Coset diagrams defined for the transitive actions of PSL(2, Z) on projective line over a Galois field F _p , PL(F _p ), where p is prime, are used to obtain a formula for finding the number of subgroups of index p + 1 of the modular group PSL(2, Z). Some intransitive actions of PSL(2, Z) on PL(F _p ) for some special values of θ, when θ ∈ F _p , are also studied.     

Polynomial Rings Over Weak Armendariz Rings need not be Weak Armendariz

It is proved that there exists a weak Armendriz ring R over which the polynomial ring R[x] is not weak Armendariz. This answers an open question of Liu and Zhao in 2006 in the negative, and eliminates the misunderstanding of the question having a positive solution by Hashemi in 2008.     

The McCoy Condition on Noncommutative Rings

We introduce the multiplication algebra of a Bernstein algebra, establish its Peirce decomposition relative to an idempotent of A and state some basic properties of this algebra of endomorphtsms     

A note on liftings of modules

Let N be a zero-symmetric right near-ring with identity. In 1993, S. Bagley introduced a construction for N[x], the near-ring of polynomials with coefficients from N. In this paper we study the central elements of N[x], C(N[x]), and we characterize C(N[x]) in terms of C(N) for a class of near-rings. We also introduce a new generalization for the center of a ring to the near-ring case, and we show that this new generalization yields a near-ring which properly contains C(N[x]) for a certain class of near-rings N.     

Degree 2 Polynomial Maps from Spheres to the Two Sphere Over Any Field

We classify all polynomial maps of degree at most 2 from the n-sphere to the 2-sphere over any field.     

Identities of Group Algebras

We consider polynomial identities of group algebras over a field F of characteristic zero. We prove that any PI group algebra satisfies the same identities as a matrix algebra M _n (F ), where n is the maximal degree of finite dimensional representations of the group over algebraic extensions of F.     

Structure Theorems of Mixable Shuffle Algebras

Mixable shuffle algebras are generalizations of the well-known shuffle algebra and quasi-shuffle algebra with broad applications. In this article we study the ring theoretic structures of mixable shuffle algebras with coefficients in a field motivated by the well-known work of Radford that a shuffle algebra with rational coefficients is a polynomial algebra in Lyndon words. To consider coefficients in a field of positive characteristic p, we carefully study the Lyndon words and their p-variations. As a result, we determine the structures of a quite large class of mixable shuffle algebras by providing explicit sets of generators and relations.     

S-Noetherian Rings of the Forms 𝒜[X] and 𝒜[[X]]

Let 𝒜 = (A _n ) _n≥0 be an ascending chain of commutative rings with identity, S ? A ₀ a multiplicative set of A ₀, and let 𝒜[X] (respectively, 𝒜[[X]]) be the ring of polynomials (respectively, power series) with coefficient of degree i in A _i for each i ∈ ?. In this paper, we give necessary and sufficient conditions for the rings 𝒜[X] and 𝒜[[X]] to be S ? Noetherian.