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The Interplay Between Topological Algebras Theory and Algebras of Holomorphic Functions
Published in Michael Ruzhansky, Hemen Dutta, Advanced Topics in Mathematical Analysis, 2019
Theorem 17.3.12 (Frisch 1965 Frisch (1965), Siu 1969 Siu (1969)). If K is a semianalytic compact set which has a fundamental system of Stein neighbourhoods in a complex space X, thenO(K)is a noetherian ring.
Integral Domains, Ideals, and Unique Factorization
Published in Richard A. Mollin, Algebraic Number Theory Second, 2011
1.32. If R is a Noetherian ring, prove that any finitely generated R-module is Noetherian.
On the finiteness of accessibility test for nonlinear discrete-time systems
Published in International Journal of Control, 2021
Mohammad Amin Sarafrazi, Ewa Pawłuszewicz, Zbigniew Bartosiewicz, Ülle Kotta
Since is a Noetherian ring, from the Hilbert Basis Theorem (Cox, Little, & OShea, 1992), it follows that there exists some κ such that . Now, from the proof of Theorem 4.1 . From the fact that (see Cox et al., 1992), definition of sets and Theorem 4.1 it follows that . Hence, . Then from we obtain , which using Theorem 4.1 gives . Since is the smallest integer such that , therefore from we have .
On the Primary Decomposition of k-Ideals and Fuzzy k-Ideals in Semirings
Published in Fuzzy Information and Engineering, 2021
Ram Parkash Sharma, Madhu Dadhwal, Richa Sharma, S. Kar
Ideals play an important role in both ring theory and semiring theory. But in the absence of additive inverses in semirings, the structure of ideals in semirings differs from that of ring theory. The ideals in semirings possessing the very obvious property of ideals of rings, ‘if , , then ’ are known as k-ideals and the role of these ideals in semirings becomes significant in the absence of additive inverses. The results which are true for ideals in rings have also been established for k-ideals in semirings by various authors (cf. [1–13]). In view of these facts various researchers attempted the primary decomposition for k-ideals in semirings analogous to the primary decomposition of ideals in rings: In a commutative Noetherian ring, every ideal can be decomposed as a finite intersection of primary ideals (Lasker–Noether Theorem [14]). For a deep study of primary ideals in rings and semirings one can refer to [8–12, 15–19].
Some remarks on global analytic planar vector fields possessing an invariant analytic set
Published in Dynamical Systems, 2021
The Nullstellensatz for the ring of real analytic functions germs is well known from Risler [11], that is, if is the Noetherian ring of real analytic functions at then (2) and (3) hold.