DeMeyer, "
Zero divisor graphs of semigroups," Journal of Algebra, vol.
Let a hyperideal I of R be [delta]-primary and then every
zero divisor of the quotient hyperring R/I is [delta] nilpotent.
An element x [member of] NQR is called a
zero divisor if there exists a nonzero element y [member of] NQR such that xy = 0.
Now suppose that ([a.sub.1], [b.sub.1]) [member of] [S.sup.2] is a
zero divisor, so there exists a nonzero ([a.sub.2], [b.sub.2]) [member of] [S.sup.2], such that ([a.sub.1], [b.sub.1]) [cross product] ([a.sub.2], [b.sub.2]) = (0,0).
Wickham, "Local rings with genus two
zero divisor graph," Communications in Algebra, vol.
For convenience, the following notation is used: for a given matrix M [member of] [R.sup.sxt], the right
zero divisor of M is denoted by [S.sub.M], which satisfies M[S.sub.M] = 0 with rank [S.sub.M] = t - rank M.
where: [(M).sup.~L.sub.r,m] ([(M).sup.~R.sub.n,r])--left (right) canonizator of matrix M; [[bar.M].sup.L.sub.m-r,n] ([[bar.M].sup.R.sub.n,n-r])--left (right)
zero divisor of the matrix m; r = rank(M); [I.sub.r]--identity matrix of size r x r.
Coverage includes a guide to closure operations in commutative algebra, a survey of test ideals, finite-dimensional vector spaces with Frobenius action, finiteness and homological conditions in commutative group rings, regular pullbacks, noetherian rings without finite normalization, Krull dimension of polynomial and power series rings, the projective line over the integers, on
zero divisor graphs, and a closer look at non-unique factorization via atomic decay and strong atoms.
Lee, Power series rings satisfying a
zero divisor porperty, Comm.
An element a [member of] R\{0} is a S-weak
zero divisor if there exists b [member of] R\{0, a} such that a, b = 0 satisfying the following conditions: There exists x, y [member of] R\{0, a, b} such that
Now consider non-differential
zero divisors. These type of quantities are distinct elements of the algebra and thus in physical applications could be corresponded to the unit signals (elementary particles).