Existence of non-standard ideals in the formal power series algebra $$\pmb {{\mathfrak {F}}({\mathbb {Z}}_{+}^{2}; {\mathbb {C}})}$$ F ( Z + 2 ; C )

H. V. Dedania () and K. R. Baleviya ()
Additional contact information
H. V. Dedania: Sardar Patel University
K. R. Baleviya: Sardar Patel University

Indian Journal of Pure and Applied Mathematics, 2024, vol. 55, issue 4, 1160-1165

Abstract: Abstract This article serves four purposes: (1) A complete characterization of semigroup ideals in $${\mathbb {Z}}_{+}^{2}$$ Z + 2 is given; (2) The concept of standard ideals is defined in most general set up; (3) Unlike the algebra $${\mathfrak {F}}({\mathbb {Z}}_{+};{\mathbb {C}})$$ F ( Z + ; C ) , there is a non-standard ideal in the algebra $${\mathfrak {F}}({\mathbb {Z}}_{+}^{2};{\mathbb {C}})$$ F ( Z + 2 ; C ) ; and (4) This is a first step in the direction of studying standard closed ideals in $$\ell ^1({{\mathbb {Z}}}_+^2, \, \omega )$$ ℓ 1 ( Z + 2 , ω ) . It is also proved that, under a certain condition on $$f\in {\mathfrak {F}}({\mathbb {Z}}_{+}^{2};{\mathbb {C}})$$ f ∈ F ( Z + 2 ; C ) , the ideal $$I_{f}=f *{\mathfrak {F}}({\mathbb {Z}}_{+}^{2};{\mathbb {C}})$$ I f = f ∗ F ( Z + 2 ; C ) is always a standard ideal. Though the proofs are elementary, the results will give more clarity about standard ideals in the formal power series algebras.

Keywords: Semigroup; Semigroup ideal; Algebra; Standard ideals; And Non-standard ideals; Primary 16D25; Secondary 20M12; 46H10 (search for similar items in EconPapers)
Date: 2024
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s13226-023-00416-z Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:indpam:v:55:y:2024:i:4:d:10.1007_s13226-023-00416-z

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/13226

DOI: 10.1007/s13226-023-00416-z

Access Statistics for this article

Indian Journal of Pure and Applied Mathematics is currently edited by Nidhi Chandhoke

More articles in Indian Journal of Pure and Applied Mathematics from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().