Lemma 1
Posted: Thu May 26, 2016 1:31 pm
Let \(\displaystyle{\left(A,+,\cdot\right)}\) be a commutative ring with unity and \(\displaystyle{S\subseteq A}\) a
multiplicative subset of \(\displaystyle{A}\). Consider the natural homomorphism \(\displaystyle{i:A\to S^{-1}\,A}\).
Let \(\displaystyle{J}\) be an ideal of \(\displaystyle{\left(A,+,\cdot\right)}\). Then,
\(\displaystyle{J^{e}=i(J)\,(S^{-1}\,A)=\left\{\sum_{k=1}^{n}i(a_k)\,\dfrac{b_k}{s_k}: a_k\in J\,,b_k\in A\,,s_k\in S\,,k\in\mathbb{N}\right\}}\)
is an ideal of \(\displaystyle{S^{-1}\,A}\). Prove that
\(\displaystyle{J^{e}=\left\{\dfrac{r}{s}: r\in J\,\,,s\in S\right\}}\).
multiplicative subset of \(\displaystyle{A}\). Consider the natural homomorphism \(\displaystyle{i:A\to S^{-1}\,A}\).
Let \(\displaystyle{J}\) be an ideal of \(\displaystyle{\left(A,+,\cdot\right)}\). Then,
\(\displaystyle{J^{e}=i(J)\,(S^{-1}\,A)=\left\{\sum_{k=1}^{n}i(a_k)\,\dfrac{b_k}{s_k}: a_k\in J\,,b_k\in A\,,s_k\in S\,,k\in\mathbb{N}\right\}}\)
is an ideal of \(\displaystyle{S^{-1}\,A}\). Prove that
\(\displaystyle{J^{e}=\left\{\dfrac{r}{s}: r\in J\,\,,s\in S\right\}}\).