Let $R$ be a ring, $S$ is a multiplicative subset of $R$. $a$ is an arbitrary element of $S$. Should there be 2 element $b,c \in S, b, c \neq 1$ such that $a=b.c$?
If not please give a counter example. Thank you very much!
Let $R$ be a ring, $S$ is a multiplicative subset of $R$. $a$ is an arbitrary element of $S$. Should there be 2 element $b,c \in S, b, c \neq 1$ such that $a=b.c$?
If not please give a counter example. Thank you very much!
No, there is no such requirement. Consider $R=\mathbb{Z}$ and let $S$ be the set of all positive integers not divisible by $2$. This is a multiplicative set, and contains $3$. However, any expression of $3$ as a product $3=bc$ with $b$ and $c$ positive integers will have $b=1$ or $c=1$.
Consider $R=\Bbb Z$ and $S=a\Bbb Z$ with $a\not\in\{-1,0,1\}$, especially in light of prime factorization.
More specifically, say we take $R$ to be the integers and the multiples of some nonzero nonunit $a$ to be our multiplicative subset $S$. For a quick example, let $a=2$ and so $S$ is the set of even numbers closed under multiplication. Can $2$ be written as the product of two even numbers?