# Algebra: Some Recent Advances by K. T. Arasu, Surinder K. Sehgal (auth.), I. B. S. Passi By K. T. Arasu, Surinder K. Sehgal (auth.), I. B. S. Passi (eds.)

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This is the so-called cancellation question. Addressing this we have the following fundamental theorem due to Bass [B 2] proved in 1964. 2. (Bass) Let R be a ring offinite Krull dimension d and let P be a projective R-module of rank ~ d + 1. Then P is cancellative. 2) are the best possible in the general setup. 3. Let R denote the field of real numbers and let R = R[X, Y, Z]/(X 2 + y2 + Z2 -1) denote the coordinate ring of the reaI2-sphere. Then R is a UFD of dimension 2. Letx, y, z denote the images of X, Y, Z respectively in R.

2nd Ed. Wiley, New York, 1986. Hall, M. , Cyclic incidence matricess. Canad. J. , 3, 495-502, 1951. , Computer investigation of difference sets. Math. , 19,73-78,1965. , On bounds for groups of multipliers of planar difference sets. J. Algebra, 148, 325-336, 1992. Ho, C. , On multiplier groups of finite cyclic planes. J. Algebra, 122, 250-259, 1989. , Planar Singer groups with even order multiplier groups. In: Finite Geometry and Combinatorics. (Eds. ,) Cambridge University Press, Cambridge, 187-198, 1993.

The group algebras for which U(KG) satisfies a semigroup identity are studied. Recently in Giambruno et al.. (1997) Hartley's conjecture has been confirmed for an infinite field K. 7. (Passman, 1997) Let K be an infinite field ofcharacteristic p > and G a torsion group. Then the following are equivalent:- ° (i) U(KG) satisfies a group identity. (ii) G has a normal p-Abelian subgroup offinite index and G' is a p-group of bounded period. (iii) U(KG) satisfies the group identity (x, y)pk (x, y) = x-Iy-I xy .