Existence of non-standard ideals in the formal power series algebra F(Z+2;C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pmb {{\mathfrak {F}}({\mathbb {Z}}_{+}^{2}; {\mathbb {C}})}$$\end{document}

This article serves four purposes: (1) A complete characterization of semigroup ideals in Z+2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}_{+}^{2}$$\end{document} is given; (2) The concept of standard ideals is defined in most general set up; (3) Unlike the algebra F(Z+;C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {F}}({\mathbb {Z}}_{+};{\mathbb {C}})$$\end{document}, there is a non-standard ideal in the algebra F(Z+2;C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {F}}({\mathbb {Z}}_{+}^{2};{\mathbb {C}})$$\end{document}; and (4) This is a first step in the direction of studying standard closed ideals in ℓ1(Z+2,ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^1({{\mathbb {Z}}}_+^2, \, \omega )$$\end{document}. It is also proved that, under a certain condition on f∈F(Z+2;C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in {\mathfrak {F}}({\mathbb {Z}}_{+}^{2};{\mathbb {C}})$$\end{document}, the ideal If=f∗F(Z+2;C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{f}=f *{\mathfrak {F}}({\mathbb {Z}}_{+}^{2};{\mathbb {C}})$$\end{document} 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.