Groups generated by a set and Ring generated by a set

Question

I was studying Ring and modules and now I am studying Group theory. I feel very confuse regarding generating set . In Ring Theory if we have an Ideal $ \ I = < x,y > $ Then x doesn't contained in that Ideal, however in Group theory if we have $ \ G = < x, y >$ then x is in the group. Can any one please show how can we write x as a product of the generators.

Answer 1

The right way to think of the subgroup $\langle X \rangle$ of a group $G$ generated by a subset $X$ of $G$ is that it is the smallest subgroup of $G$ that contains $X$. Or more formally, it is the intersection of all subgroups of $G$ that contain $X$. From that point of view, it is clear that $X \subseteq \langle X \rangle$ (because there is at least one subgroup of $G$, namely $G$ itself, that does contain $X$). Given this definition of $\langle X \rangle$, we can show that $\langle X \rangle$ comprises all elements of $G$ that can be written as products of elements of $X$ or their inverses, where, if $x \in X$, we view $x$ itself as a product of 1 element, namely $x$ itself.

The above description generalises to other situations with "products of elements" replaced by the appropriate algebraic operations: e.g., the ideal generated by a subset $X$ of a ring $R$ is the smallest ideal of $R$ containing $X$ and comprises all the elements of $R$ you can obtain from elements of $X$ by addition and by multiplication by arbitrary elements of $R$.

Answer 2

The confusion stems from the definition of “ideal generated by $x$” (let's keep it simple, for the moment, with $R$ a commutative ring).

There are two possible definitions:

1. the ideal generated by $x$ is $Rx$
2. the ideal generated by $x$ is the intersection of all the ideals containing $x$

These two objects can differ if the ring doesn't have an identity. For instance, if $R=2\mathbb{Z}$ and $x=4$, then $$ Rx=\{0,8,-8,16,-16,\dotsc\} $$ indeed doesn't contain $4$; on the other hand, the intersection of all ideals containing $4$ is $\{0,4,-4,8,-8,12,-12,16,-16,\dotsc\}$ because this is clearly an ideal and every ideal containing $4$ must contain $4n$, for every integer $n$.

Without a clear statement of the definition of the ideal $\langle x,y\rangle$, not much more can be said. If $\langle x,y\rangle$ is assumed to be $Rx+Ry$, then this does not necessarily contain $x$ or $y$. If it is the smallest ideal containing $x$ and $y$, then it obviously contains both $x$ and $y$.

As far as I know, the second definition (smallest ideal containing the given elements) is the one commonly used.