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.

The most remote sources of partial actions of Hopf algebras can be found in the theory of partial Galois extensions, which was a generalization of the Galois theory for

commutative rings by Chase, Harrison and Rosenberg to the case of partial group actions.

The Kipnis and Hibshoosh scheme [61] that is based on

commutative rings is subject to known plaintext key-recovery attack [97-98].

Brown, Matrices Over

Commutative Rings, Marcel Dekker, New York, NY, USA, 1993.

1] was through the scheme theory developed by Deitmar [1], which is based on the observation that

commutative rings over [F.

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.

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.

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.

On strong orthogonal systems and weak permutation polynimials over finite

commutative rings, Finite Fields Appl.

The complexity for equivalence for

commutative rings.

Finally, in Section 5, we extend the notion of the universal side divisors of

commutative rings to semimodules and show that every Euclidean semimodule contains a universal side divisor which consequently implies that a cyclic semimodule with no universal side divisors is never Euclidean.

Chapters address group theory,

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