Let be a field, and the set of monomials of . It is well known that the set of monomial ideals of is in a bijective correspondence with the set of all subsemiflows of the -semiflow . We generalize this to the case of term ideals of , where is a commutative Noetherian ring. A term ideal of is an ideal of generated by a family of terms , where and are integers .
Let be a commutative ring with identity. If a ring is contained in an arbitrary union of rings, then is contained in one of them under various conditions. Similarly, if an arbitrary intersection of rings is contained in , then contains one of them under various conditions.
Nous introduisons pour les systèmes linéaires constants les reconstructeurs intégraux et les correcteurs proportionnels-intégraux généralisés, qui permettent d’éviter le terme dérivé du PID classique et, plus généralement, les observateurs asymptotiques usuels. Notre approche, de nature essentiellement algébrique, fait appel à la théorie des modules et au calcul opérationnel de Mikusiński. Plusieurs exemples sont examinés.
For constant linear systems we are introducing integral reconstructors and generalized proportional-integral controllers, which permit to bypass the derivative term in the classic PID controllers and more generally the usual asymptotic observers. Our approach, which is mainly of algebraic flavour, is based on the module-theoretic framework for linear systems and on operational calculus in Mikusiński's setting. Several examples are discussed.
Rings of formal power series with exponents in a cyclically ordered group were defined in [2]. Now, there exists a “valuation” on : for every in and in , we let be the first element of the support of which is greater than or equal to . Structures with such a valuation can be called cyclically valued rings. Others examples of cyclically valued rings are obtained by “twisting” the multiplication in . We prove that a cyclically valued ring is a subring of a power series ring with...
In this paper we first calculate the number of vertices and edges of the intersection graph of ideals of direct product of rings and fields. Then we study Eulerianity and Hamiltonicity in the intersection graph of ideals of direct product of commutative rings.
Order complex is an important object associated to a partially ordered set. Following a suggestion from V. A. Vassiliev (1994), we investigate an order complex associated to the partially ordered set of nontrivial ideals in a commutative ring with identity. We determine the homotopy type of the geometric realization for the order complex associated to a general commutative ring with identity. We show that this complex is contractible except for semilocal rings with trivial Jacobson radical when...
In this paper, we use Zorn’s Lemma, multiplicatively closed subsets and saturated closed subsets for the following two topics: (i) The existence of prime submodules in some cases, (ii) The proof that submodules with a certain property satisfy the radical formula. We also give a partial characterization of a submodule of a projective module which satisfies the prime property.