Among their topics are isonoetherian and isoartinian modules, the universal

abelian regular ring, a characterization of t-tilting finite algebras, multisorted modules and their module theory, pure projective modules over non singular serial rings, and intrinsic valuation entropy.

Our paper [OO] is a sequel to a series by the first author [Od14, Od18], which compactified both the moduli space of compact Riemann surfaces [M.sub.g](g [greater than or equal to] 2) and that of principally polarized

abelian varieties [A.sub.g].

On products of cyclic and

abelian finite p-groups (p odd) ...

Furthermore, we have demonstrated the existence of such (i.e., (anti-)co-BRST) symmetries in the cases of

Abelian p-form (p = 1,2,3) gauge theories in the two (1 + 1) dimensions, four (3+1) dimensions, and six (5 + 1) dimensions of spacetime (see, e.g., [18] and references therein).

Dhompongsa [5] proved that each infinite dimensional unital

Abelian real Banach algebra X with [OMEGA](X) [not equal to] 0 satisfying (i) if x, y [member of] X is such that [absolute value of [tau](x)] [less than or equal to] [absolute value of [tau](y)] for each [tau] [member of] [OMEGA](X) then [parallel]x[parallel] [less than or equal to] [parallel]y[parallel] and (ii) inf {r(x) : x [member of] X, [parallel]x[parallel] = 1} > 0 does not have the fixed point property.

Abelian anyons have been detected and play a major role in the fractional quantum Hall effect.

Let [C.sub.r](A) be the free

abelian group generated by the set [[DELTA].sub.r](A) consisting of all elements of the form ([a.sub.1],...,[a.sub.r],[[gamma].sub.1],...,[[gamma].sub.r]) where [a.sub.1],...,[a.sub.r] are pairwise distinct elements of A, and [[gamma].sub.1],...,[[gamma].sub.r] are any elements of [GAMMA]; in particular, [C.sub.r](A) = Z if r = 0.

More precisely, for a field k, a joint determinant D (= [D.sub.l]) (l [greater than or equal to] 1) is defined as a map from the set of l-tuples of commuting matrices in [GL.sub.n](k) (n [greater than or equal to] 1) into some

abelian group (G, +) which satisfies the following properties.

Let H be a finite

abelian group written additively and End(H) be the endomorphism ring of H.

It is well known that all endomorphisms of an

Abelian group form a ring and many of its properties can be characterized by this ring.

These results on Euclidean self-dual cyclic codes have been generalized to

abelian codes in group algebras [6] and the complete characterization and enumeration of Euclidean self-dual

abelian codes in principal ideal group algebras (PIGAs) have been established.