Edmund Puczyłowski

On maximal ideals and the Brown-McCoy radical of polynomial rings

In this paper we obtain some results on maximal ideals of polynomial rings R[x] in one indeterminate x. In particular we complete Ferreros characterization [5] of rings R with an identity such that R[x] contains a maximal R-disjoint ideal, i.e., a maximal ideal M satisfying M n R = 0. We also get several results on the Brown-McCoy radical of R[x]. Recall that for a given ring A the Brown-McCoy radical U(A) of A is defined as the intersection of all ideals I of A such that A/I is a simple ring with an identity. In particular a ring is Brown-McCoy radical if and only if it cannot be homomorphically mapped onto a ring with an identity, or equivalently, onto a simple ring with an identity. In [7] Krernpa proved that for every ring R, U(R[xJ) = (U(R[x]) n R)[xJ. We shall show that U(R[x]) n R is equal to the intersection of all prime ideals I of R such that the centre of R/I has a non-zero intersection with each non-zero ideal of RII. In particular, if R is a nil ring, then R[x] is Brown-McCoy radical, i.e., R[x] cannot be homomorphically mapped onto

Quasi-duo skew polynomial rings

A characterization of right (left) quasi-duo skew polynomial rings of endomorphism type and skew Laurent polynomial rings are given. In particular, it is shown that (1) the polynomial ring R[x] is right quasi-duo iff R[x] is commutative modulo its Jacobson radical iff R[x] is left quasi-duo, (2) the skew Laurent polynomial ring is right quasi-duo iff it is left quasi-duo. These extend some known results concerning a description of quasi-duo polynomial rings and give a partial answer to the question posed by Lam and Dugas whether right quasi-duo rings are left quasi-duo.

Hereditariness of strong and stable radicals

The aim of this paper is to discuss some relations among hereditary, strong and stable radicals. In particular we investigate hereditariness of lower strong and stable radicals. Some facts obtained are related to some results and questions of [2, 6, 7]. All rings in the paper are associative. Fundamental definitions and properties of radicals may be found in [9]. Definitions of hereditary and strong radicals are used as in Sands [7]. We say that a radical S is left (right) stable if (ρ): for every ring R and every left (right) ideal I of R it follows S(I) ⊆ S(R) .

A polynomial ring that is Jacobson radical and not nil

In [1] Amitsur conjectured that if a polynomial ring in one indeterminate is Jacobson radical then it is a nil ring. We shall construct an example disproving this conjecture.

RINGS WHICH ARE SUMS OF TWO SUBRINGS SATISFYING POLYNOMIAL IDENTITIES

*The author was supported by KBN Grant 2 PO3A 039 14.

Rings which are sums of two subrings

Abstract The main results concern radicals and polynomial identities of rings which are sums of two subrings. It is proved that a ring which is a sum of a nil subring of bounded index and a ring satisfying a polynomial identity also satisfies a polynomial identity. Filds which are sums of two Jacobson radical subrings are classified. Several open questions are answered.

THE SINGULAR IDEAL AND RADICALS

Some properties of the singular ideal are established. In particular its behaviour when passing to one-sided ideals is studied. Obtained results are applied to study some radicals related to the singular ideal. In particular a radical S such that for every ring R, S.R/ and R=S.R/ are close to being a singular ring and a non-singular ring, respectively, is constructed.

Nil algebras with nonradical tensor square

The paper contains some simple observations on the tensor square of algebras. Applied to the well-known Golod examples, they allow us to pro- duce nil algebras with nonradical tensor square.

On finiteness conditions of modular lattices

In [Gd] Goodearl proved that if for every essential submodule N of a module M M/N is a Noetherian module, then the module M/SocM is Noetherian. Then in [AS], Al-Khazzi and Smith got that if every small submodule of a module M is Artinian then so is the Jacobson radical 𝔍(M) of M. These results are dual to each other in the lattice theory sense ( recall that is essential submodule of M}). However the proofs in [AS] and [Gd] are not dual at all. Hence it is natural to ask whether there is a common proof of the both results. The best it would be to extend one of these results to complete modular lattices or to a satisfactory subclass of such lattices. Then to get any of the results it would be enough to take the lattice of submodules of a module or its dual. Attempts to find such an extension inspired our studies in this paper. We did not settle the general problem but obtained such an extension in cases of complete modular lattices which are upper continuous, lower continuous or distributive. Moreover we go...

Prime ideals of graded rings and related matters

Let R be a ring graded by an abelian group.We study prime ideals of R that are maximal for not containing nonzero homogeneous elements.Also prime ideals of the symmetric graded Martindale ring of quotients of R are investigated.The results are applied to study when R is a Jacobson ring in case R is a Z-graded ring or a group ring of a finitely generated abelian group, or in case R is right Noetherian and strongly graded by a polycyclic-by-finite group.