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
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 , Proposition 2.6).
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.