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.

Treating electric flux lines as the dynamical degrees of freedom of gauge theories leads to a representation of Wilson loop associated with a closed curve C for

Abelian gauge theories.

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.