let R be a commutatuve ring that does not have unity. For a fixed a in R, prove that the set (a)={na+ra|n in Z, r in R} is an ideal of R that contains the element a.
let R be a commutatuve ring that does not have unity. For a fixed a in R, prove that the set (a)={na+ra|n in Z, r in R} is an ideal of R that contains the element a.
This is pretty plug and chug. What have you tried?
let R be a commutatuve ring that does not have unity. For a fixed a in R, prove that the set (a)={na+ra|n in Z, r in R} is an ideal of R that contains the element a.
In case you are interested in some additional details,
If a is in the center of R (assuming R is not necessary commutative), then your (a)={na+ra|n in Z, r in R} is an ideal of R containing a and contained in every ideal containing a. (Hungerford "Algebra", p 124)
Last edited by aliceinwonderland; Feb 22nd 2010 at 11:58 PM.