NQR(Z) the neutrosophic quadruple ring of integers is a neutrosophic quadruple

integral domain.

In [6], it is shown that an

integral domain with no universal side divisors can not be Euclidean.

The proof of this lemma uses (2, Theorem (2.10)) stating that given any

integral domain R and a matrix X of variables, the ideal generated by the maximal minors of X in the polynomial ring R[X] is prime.

(Conversely, a nearly limitless family of hash functions can be had, each member of which is identified with a different choice of T.) Second, the polynomial [x.sup.w] + 1 is composite, so that the ring is not an

integral domain. As the Appendix points out (Section A.5), for most implementations the lack of an

integral domain here does not matter.

Chapters four through seven cover abstract groups and monoids, orthogonal groups, stochastic matrices, Lagrange's theorem, groups of units of monoids, homomorphisms, rings, and

integral domains. The first seven chapters provide basic coverage of abstract algebra, suitable for a one-semester or two-quarter course.

This is followed by a range of current topics such as multilinear algebra, matrix polynomials and equations, perturbation theory, pseudospectra, inverse problems,

integral domains, and spectral sets, among many others.

Some simple considerations about symmetry properties of the

integral domains T in (2)-(4) deserve to be better investigated.

This comprehensive treatment of an under-studied aspect of commutative algebra describes various simple extensions and their properties, in particular properties of simple ring extensions of Noetherian

integral domains. Oda (mathematics, Kochi U.) and Yoshida (mathematics, Okayama U.

They then discuss integers, rings and ordered

integral domains. They also include an elementary proof of the Fundamental Theorem of Algebra.

Among the topics are greatest common divisors, integer multiples and exponents, quotients of polynomial rings, divisibility and factorization in

integral domains, subgroups of cyclic groups, cosets and Lagrange's theorem, the fundamental theorem of finite abelian groups, and check digits.

Nicholson starts with a review of proofs, sets, mappings and equivalences, then in 11 chapters covers integers and permutations, groups, rings, polynomials, factorization in

integral domains, fields, modules over principal ideal domains, p-groups and the Sylow theorems, series of subgroups, Galois theory and finiteness conditions for rings and modules.

Other topics include polynomials, factorization in

integral domains, p-groups and the Sylow theorems, Galois theory, and finiteness conditions for rings and modules.