The common family of Hopf algebras for which R(H) is a

commutative algebra consists of quasitriangular Hopf algebras introduced by Drinfel'd [9] in the context of quantum groups.

Then section 5 deals with an algebraic structure that gathers all the quadrangulations together, namely a

commutative algebra endowed with a derivation.

Eisenbud,

Commutative Algebra, with a view toward Algebraic Geometry.

Besides that, we introduce here also another system of Lax equations for a more general deformation of the basic generator of the

commutative algebra.

Readers should have a background in number theory,

commutative algebra, and the general theory of schemes.

Professor Jugal Kishore Verma has made substantial and fundamental contributions to theimportant area of mixed multiplicities in

commutative algebra.

We have been teaching a mathematics course in

Commutative Algebra and Algebraic Geometry at Simon Fraser University since 2006.

There are important theorems that are commonly treated in advanced books on

commutative algebra without proofs, because the existing proofs are so difficult, say Majadas and Rodicio (both U.

A

commutative algebra X with product x [o] y is called a Jordan algebra if [x.

For the next result we need to recall some concepts from

commutative algebra.

A Term of

Commutative Algebra by Allan Altman and Steve Kleiman is available in digital and print format; the digital version is available to students free of charge, and the book has been adopted at MIT for the 2012-2013 academic year.

This text introduces the theory of separable algebras over commutative rings, covering background on rings, modules, and

commutative algebra, then the key roles of separable algebras, including Azumaya algebras, the henselization of local rings, and Galois theory, as well as AaAaAeAa[umlaut]ta algebras, connections between the theory of separable algebras and Morita theory, the theory of faithfully flat descent, cohomology, derivations, differentials, reflexive lattices, maximal orders, and class groups.