Principal ideal in a ring without identity

(Hungerford exercise 31, page 143)

Let R be a commutative ring without identity and let $\displaystyle a \in R $

Show that $\displaystyle A = \{ ra + na \ | \ r \in R, n \in \mathbb{Z} \} $ is an ideal containing a and that every ideal containing a also contains A. (A is called the prinicipal ideal generated by a)

Re: Principal ideal in a ring without identity

For the first one: take $\displaystyle r=0, n=1$

For the second one: In an ideal $\displaystyle I$, multiplication by elements in $\displaystyle R$ and addition by elements in $\displaystyle I$ produces elements in $\displaystyle I$. The result should now be obvious (though you should prove it more rigorously, shouldn't be more than a few lines).