# Principal ideal in a ring without identity

• March 15th 2013, 06:45 AM
Bernhard
Principal ideal in a ring without identity
(Hungerford exercise 31, page 143)

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

Show that $A = \{ ra + na \ | \ r \in R, n \in \mathbb{Z} \}$ is an ideal containing a and that every ideal containing a also contains A. (A is called the prinicipal ideal generated by a)
• March 15th 2013, 05:27 PM
Gusbob
Re: Principal ideal in a ring without identity
For the first one: take $r=0, n=1$

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