The divisibility theory of

commutative rings is a fundamental and persisting topic in mathematics that entails two main aspects: determining irreducibility and finding a factorization of the reducible elements in the ring.

He walks readers who have a reasonable acquaintance with the more elementary facts concerning

commutative rings and modules through most of the known results on formal groups, and applications that do not require too much extra apparatus.

Smith defined weakly prime ideals in

commutative rings, an ideal P of a ring R is weakly prime if 0 [not equal to] ab [member of] P implies a [member of] P or b [member of] P.

Chapters address group theory,

commutative rings, Galois theory, noncommutative rings, representation theory, advanced linear algebra, and homology.

For one, by building on the well-studied setting of modules over

commutative rings, we get a theory where the considerable power and development of commutative algebra can be easily brought to bear.

They also include graphs for algebraic structures like commutative semigroups, loops, communicative groupoids, and

commutative rings.

Chapters consider local and reduced rings, and

commutative rings in general, as well as the classification of minimal ring extensions, linear systems theory over

commutative rings, and the history and summary of asymptotic stability of associated or attached prime ideals.

The 64 papers in this collection explore field theory and polynomials,

commutative rings and algebras, matrix theory, associative rings, K-theory, group theory and generalizations, topological groups, Lie groups, and differential geometry.

Fourteen papers accepted for the December 2006 conference describe the structure of incidence rings of group automata, apply diagram categories from statistical mechanics to representation theory, and examine determinantal and Pfaffian ideals of symmetric matrices over general

commutative rings.

Brzezinski (University of Wales) and Wisbauer (Heinrich Heine University) develop the theory of coalgebras over

commutative rings and their comodules, and present known results on the structure of corings.

Arithmetic properties of

commutative rings and monoids.

Hirano, On annihilator ideals of a polynomial ring over a non

commutative ring, J.