(Hungerford exercise 31, page 143)

Let R be a commutative ring without identity and let

Show that is an ideal containing a and that every ideal containing a also contains A. (A is called the prinicipal ideal generated by a)

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- Mar 15th 2013, 07:45 AMBernhardPrincipal ideal in a ring without identity
(Hungerford exercise 31, page 143)

Let R be a commutative ring without identity and let

Show that is an ideal containing a and that every ideal containing a also contains A. (A is called the prinicipal ideal generated by a) - Mar 15th 2013, 06:27 PMGusbobRe: Principal ideal in a ring without identity
For the first one: take

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