Projective prime ideals and localisation in pi-rings

The results here generalise [2, Proposition 4.3] and [9, Theorem 5.11]. We shall prove the following. THEOREM A. Let R be a Noetherian PI-ring. Let P be a non-idempotent prime ideal of R such that PR is projective. Then P is left localisable and RP is a prime principal left and right ideal ring. We also have the following theorem. THEOREM B. Let R be a Noetherian PI-ring. Let M be a non-idempotent maximal ideal of R such that MR is projective. Then M has the left AR-property and M contains a right regular element of R

Smooth PI algebras with finite divisor class group

We have shown in an earlier paper that the divisor class group of the centre of a smooth PI algebra with trivial K(0) is a torsion group of finite exponent. We show here that this group need not be finite even in the affine case. Our example is an Azumaya algebra of global dimension 2. We also provide a positive result in a special case

Removal of Copper by Adsorption on Fly Ash

564-573Describes the studies on removal of copper from mixed metal solution of Cu, Cd, Mn, Ni and Zn by adsorption on fly ash at room temperature. The removal of copper is observed to increase with contact time. The adsorption on fly ash follows Langmuir adsorption isotherm. Maximum percentage of copper removal is observed at a pH of 5.0. The results show more than 75 per cent removal of copper

The failure of the Artin–Rees property for the Jacobson radical in prime Noetherian rings

We provide an example of a prime Noetherian ring whose Jacobson radical fails to satisfy the Artin-Rees (AR) property on either side. The ring constructed is a finite module over its Noetherian centre. This settles a longstanding question