Prime radicals of constants of algebraic derivations

Let R be a prime ring with extended centroid F and let δ be an F-algebraic continuous derivation of R with the associated inner derivation ad(b). Factorize the minimal polynomial μ(λ) of b over F into distinct irreducible factors ${\mu(\lambda)=\prod_i\pi_i(\lambda)^{n_i}}$ . Set ? to be the maximum of n _i . Let ${R^{(\delta)}{\mathop{=}\limits^{{\rm def.}}}\{x\in R\mid \delta(x)=0\}}$ be the subring of constants of δ on R. Denote the prime radical of a ring A by ${{\mathcal{P}}(A)}$ . It is shown among other things that $${\mathcal{P}}(R^{(\delta)})^{2^\ell-1}=0\quad\text{and}\quad{\mathcal{P}}(R^{(\delta)})=R^{(\delta)}\cap {\mathcal{P}}(C_R(b))$$ .     

On Litoff’s theorem

We give a short proof of Litoff’s theorem from the viewpoint of completely reducible modules.     

Right Centralizers of Semiprime Rings

Let R be a semiprime ring with Q _ml (R) the maximal left ring of quotients of R. Suppose that T: R → Q _ml (R) is an additive map satisfying T(x ²) = xT(x) for all x ∈ R. Then T is a right centralizer; that is, there exists a ∈ Q _ml (R) such that T(x) = xa for all x ∈ R.     

The Double Centralizer Theorem for Semiprime Algebras

In the paper we prove the double centralizer theorem for semiprime algebras. To be precise, let R be a closed semiprime algebra over its extended centroid F, and let A be a closed semiprime subalgebra of R, which is a finitely generated module over F. Then C _R (A) is also a closed semiprime algebra and C _R (C _R (A))?=?A. In addition, if C _R (A) satisfies a polynomial identity, then so does the whole ring R. Here, for a subset T of R, we write C _R (T):?=?{x?∈?R|xt?=?tx???t?∈?T}, the centralizer of T in R.     

On Modules Over Group Rings

Let M be a right module over a ring R and let G be a group. The set MG of all formal finite sums of the form ∑? _g?∈?G m _g g where m _g ?∈?M becomes a right module over the group ring RG under addition and scalar multiplication similar to the addition and multiplication of a group ring. In this paper, we study basic properties of the RG-module MG, and characterize module properties of (MG) _RG in terms of properties of M _R and G. Particularly, we prove the module-theoretic versions of several well-known results on group rings, including Maschke’s Theorem and the classical characterizations of right self-injective group rings and of von Neumann regular group rings.