Prime Ideals and Filters 1
نویسنده
چکیده
One can check that there exists a lattice which is non trivial, Boolean, and strict. Let X be a set. Observe that there exists a family of subsets of X which is finite and non empty. Let S be a 1-sorted structure. One can check that there exists a family of subsets of S which is finite and non empty. Let L be a non trivial upper-bounded non empty poset. Note that there exists a filter of L which is proper. We now state several propositions:
منابع مشابه
Fuzzy filters in ordered $Gamma$-semirings
We introduce the notion of ideal, prime ideal, lter, fuzzy ideal, fuzzy prime ideal, fuzzy lter of ordered $Gamma$-semiring and study their properties and relations between them. We characterize the prime ideals and lters of ordered $Gamma$-semiring with respect to fuzzy ideals and fuzzy l- ters respectively.
متن کاملZero sets in pointfree topology and strongly $z$-ideals
In this paper a particular case of z-ideals, called strongly z-ideal, is defined by introducing zero sets in pointfree topology. We study strongly z-ideals, their relation with z-ideals and the role of spatiality in this relation. For strongly z-ideals, we analyze prime ideals using the concept of zero sets. Moreover, it is proven that the intersection of all zero sets of a prime ideal of C(L),...
متن کامل$z^circ$-filters and related ideals in $C(X)$
In this article we introduce the concept of $z^circ$-filter on a topological space $X$. We study and investigate the behavior of $z^circ$-filters and compare them with corresponding ideals, namely, $z^circ$-ideals of $C(X)$, the ring of real-valued continuous functions on a completely regular Hausdorff space $X$. It is observed that $X$ is a compact space if and only if every $z^circ$-filter ...
متن کاملPrime Filters and Ideals in Distributive Lattices
The article continues the formalization of the lattice theory (as structures with two binary operations, not in terms of ordering relations). In the Mizar Mathematical Library, there are some attempts to formalize prime ideals and filters; one series of articles written as decoding [9] proven some results; we tried however to follow [21], [12], and [13]. All three were devoted to the Stone repr...
متن کاملGraded Prime Ideals Attached to a Group Graded Module
Let $G$ be a finitely generated abelian group and $M$ be a $G$-graded $A$-module. In general, $G$-associated prime ideals to $M$ may not exist. In this paper, we introduce the concept of $G$-attached prime ideals to $M$ as a generalization of $G$-associated prime ideals which gives a connection between certain $G$-prime ideals and $G$-graded modules over a (not necessarily $G$-graded Noetherian...
متن کاملI-prime ideals
In this paper, we introduce a new generalization of weakly prime ideals called $I$-prime. Suppose $R$ is a commutative ring with identity and $I$ a fixed ideal of $R$. A proper ideal $P$ of $R$ is $I$-prime if for $a, b in R$ with $ab in P-IP$ implies either $a in P$ or $b in P$. We give some characterizations of $I$-prime ideals and study some of its properties. Moreover, we give conditions ...
متن کامل