If one wants to investigate the properties and relations of homogeneous ideals in a commutative graded ring, one has as a model on the one hand the well known case of a polynomial ring and on the other hand the general commutative ideal theory. The case of a polynomial ring has been studied for the sake of algebraic geometry, and one of the methods was traditionally the passage to nonhomogeneou...

A ring of continuous functions is a ring of the form C(X), the ring of all continuous real-valued functions on a completely regular Hausdorff space X. With each ideal / of C(X), we associate certain subalgebras of C(X), and discuss their structure spaces. We give necessary and sufficient conditions for two ideals in rings of continuous functions to have homeomorphic structure spaces. Introducti...

Let K and L be compact convex sets in Rn. Suppose that, for a given dimension 1 ≤ d ≤ n − 1, every d-dimensional orthogonal projection of L contains a translate of the corresponding projection of K. Does it follow that the original set L contains a translate of K? In other words, if K can be translated to “hide behind” L from any perspective, does it follow that K can “hide inside” L? A compact...

Let $S$ be an ordered semigroup. A fuzzy subset of $S$ is anarbitrary mapping   from $S$ into $[0,1]$, where $[0,1]$ is theusual interval of real numbers. In this paper,  the concept of fuzzygeneralized bi-ideals of an ordered semigroup $S$ is introduced.Regular ordered semigroups are characterized by means of fuzzy leftideals, fuzzy right ideals and fuzzy (generalized) bi-ideals.Finally, two m...

We extend the basic fact that every ideal of a finite dimensional semisimple Lie algebra has a unique complement to the case of closed ideals of prosemisimple Lie algebras. We prove that if A is a closed ideal of a prosemisimple Lie algebra L = lim ←−−Ln (n ∈ N), where the Ln are finite dimensional semisimple Lie algebras, then there exists a unique ideal B of L such that L = A ⊕ B. Mathematics...

Let K ⊂ R d be a sufficiently round convex body (the ratio of the circumscribed ball to the inscribed ball is bounded by a constant) of a sufficiently large volume. We investigate the randomized integer convex hull I L (K) = conv(K ∩L), where L is a randomly translated and rotated copy of the integer lattice Z d. We estimate the expected number of vertices of I L (K), whose behaviour is similar...

A right ideal A of a ring R is called annihilator-small if A+ T = R; T a right ideal, implies that l(T ) = 0; where l( ) indicates the left annihilator. The sum Ar of all such right ideals turns out to be a two-sided ideal that contains the Jacobson radical and the left singular ideal, and is contained in the ideal generated by the total of the ring. The ideal Ar is studied, conditions when it ...

Given a monomial m in polynomial ring and subset L of the variables ring, principal -Borel ideal generated by is all monomials which can be obtained from successively replacing those are have smaller index. collection I = { 1 , … r } where i for (where subsets may different each ideal), we prove essence that if bipartite incidence graph among chordal bipartite, then defining equations multi-Ree...

Let R be the pseudovariety of all finite semigroups whose principal right ideals have a unique generator, let L be its dual, and let J = R∩L. Using the word problem for free pro-J semigroups, it is shown that the bilateral semidirect product Sl ∗∗ J is local, where Sl denotes the pseudovariety of all finite semilattices. The global of the pseudovariety R ∨ L is also computed and it is shown tha...