Skip to main content ☰ Contents You! < Prev ^ Up Next > \(\def\R{\mathbb{R}}
\def\C{\mathbb{C}}
\def\Q{\mathbb{Q}}
\def\N{\mathbb{N}}
\def\Z{\mathbb{Z}}
\def\mfa{\mathfrak{a}}
\def\mfb{\mathfrak{b}}
\def\mfc{\mathfrak{c}}
\def\mfp{\mathfrak{p}}
\def\mfq{\mathfrak{q}}
\def\mfm{\mathfrak{m}}
\def\ev{\mathrm{ev}}
\def\Im{\mathrm{Im}}
\def\End{\mathrm{End}}
\def\Aut{\mathrm{Aut}}
\def\Int{\mathrm{Int}}
\def\Hom{\mathrm{Hom}}
\def\Iso{\mathrm{Iso}}
\def\GL{\mathrm{GL}}
\def\ker{\mathrm{ker }}
\def\Span{\mathrm{Span}}
\def\tr{\mathrm{tr}}
\def\Rings{\mathrm{Rings}}
\def\Alg{{F\text{-Alg}}}
\def\Ann{{\rm Ann}}
\newcommand{\unit}{1\!\!1}
\usepackage{tikz}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Section 6.1 Multiplicatively closed subset
We begin with the definition of multiplicatively closed subset of a commutative ring.
Definition 6.1.1 .
A subset \(S\) of a commutative ring is said to be multiplicatively closed if following conditions are satisfied:
\(\displaystyle 1\in S\)
If \(a,b\in S\) then \(ab\in S\)
Example 6.1.2 . (Compliment of a prime ideal).
Let \(A\) be a commutative ring and let \(\mfp\) be a prime ideal in \(A\text{.}\) It is straightforward to verify that \(A\setminus\mfp\) is a multiplicatively closed subset of \(A\text{.}\)
Example 6.1.3 . (Non-zero elements of an integral domain).
Let \(A\) be an integral domain. The subset of all nonzero element of \(A\) is a multiplicatively closed set.
Example 6.1.4 .
Let \(A\) be a commutative ring and let \(a\in A\text{.}\) The set \(\{a^n:n\geq 0\}\) is a multiplicative subset of \(A\text{.}\)
