For any

commutative semiring, one has f(A) [less than or equal to] t(A) whenever A is an m byn matrix over S.

Theorem 4.1 Let B be a finite left regular band, k a

commutative ring with unit, and X,Y G A (B).

Beck, "Coloring of

commutative rings," Journal of Algebra, vol.

In constructing the course outline, faculty listed the following topics as possible topics for a first course in

commutative algebra and algebraic geometry.

This proves that the diagram is

commutative. Also, the corresponding relations are presented as

His movements may be guided principally by

commutative justice, which usually poses no knowledge problem at all.

They learned that a perfect square does not have a

commutative partner.

Since R is a

commutative unitalring, [([r.sup.2] + r).sup.2] = [r.sup.2] x [(r + 1).sup.t] = [r.sup.t] x [b.sup.t].

Every unital (

commutative or not) normed algebra (similarly, every unital p-normed algebra with p [member of] (0,1]) is a TQ-algebra, hence also a TQ-algebras (see [12], Proposition 2.6).

For

commutative semirings, needless to say, the notions of left and right Euclidean norm coincide.

In (11), Herstein proved that if R is a prime ring with char R [not equal to] 2 and R admits a non-zero derivation d such that [d(x), d(y)] = 0 for all x, y [member of] R, then R is

commutative. In (10), Filippis showed that if R be a prime ring of characteristic dierent from 2, d a non-zero derivation of R and [rho] a non-zero right ideal of R such that [[rho], [rho]][rho] [not equal to] 0 and [[d(x), x], [d(y), y]] = 0 for all x, y [member of] [rho], then d([rho])[rho] = 0.

Let N be a weak

commutative near-ring without non-zero zero divisors.