The Zykov ring of signed finite simple graphs with topological join as addition and compatible multiplication is an integral domain but not a unique factorization domain. We know that because by the graph complement operation it is isomorphic to the strong Sabidussi ring with disjoint union as addition. We prove that the Euler characteristic is a ring homomorphism from the strong ring to the in...

Let be a commutative ring with 1 and left unitary . In this papers we introduced studied concept P-small compressible (An is said to if can embedded in every of it nonzero submodule Equivalently, there exists monomorphism , retractable for non-zero homomorphism whenever as generalization respectively give some their advantages characterizations examples.

If A is a strongly noetherian graded algebra generated in degree one, then there is a canonically constructed graded ring homomorphism from A to a twisted homogeneous coordinate ring B(X,L, σ), which is surjective in large degree. This result is a key step in the study of projectively simple rings. The proof relies on some results concerning the growth of graded rings which are of independent i...

For a simply laced and hyperbolic Kac–Moody group G = G(R) over a commutative ring R with 1, we consider a map from a finite presentation of G(R) obtained by Allcock and Carbone to a representation–theoretic construction G(R) corresponding to an integrable representation V λ with dominant integral weight λ. When R = Z, we prove that this map extends to a group homomorphism ρλ,Z : G(Z)→ G(Z). We...

For every ring R, we construct a ringed space NCSpec(R) and for every ring homomorphism R → S, a morphism of ringed spaces NCSpec(S) → NCSpec(R). We show that this gives a fully faithful contravariant functor from the category of rings to a category of ringed spaces. If R is a commutative ring, we show that NCSpec(R) can be viewed as a natural completion of Spec(R). We then explain how the spac...

Entangling properties of a mixed 2-qubit system can be described by the local homogeneous unitary invariant polynomials in elements of the density matrix. The structure of the corresponding invariant polynomial ring for the special subclass of states, the so-called mixed X−states, is established. It is shown that for the X−states there is an injective ring homomorphism of the quotient ring of S...

Users’ signing order is an important factor in multisignature schemes. In fact the signing order often reflects signer’s rank in a signing group. So far, many multisignature schemes, called structured schemes, have been proposed that provide verifiability of the signing order associated with the structure of a group of signers. However, there are not many structured multisignature schemes utili...

Soft set theory, proposed by Molodtsov has been regarded as an effective mathematical tool to deal with uncertainties. Vague set is a set of objects, each of which has a grade of membership whose value is a continuous subinterval of [0, 1]. Such a set is characterized by a truth-membership function and a falsemembership function. Thus, a vague set is actually a form of fuzzy set, albeit a more ...

If A is a strongly noetherian graded algebra generated in degree one, then there is a canonically constructed graded ring homomorphism from A to a twisted homogeneous coordinate ring B(X,L, σ), which is surjective in large degree. This result is a key step in the study of projectively simple rings. The proof relies on some results concerning the growth of graded rings which are of independent i...