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  1. #1
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    Ideal

    Let R be any commutative ring and let a1,......,an be elements of R. We define
    (a1,.....,an) to be the set {r1a1+...rnan for all i, ri in R}.
    Prove that (a1,......,an) is an ideal of R. We call it the ideal generated by a1,......, an.

    I appreciate any help.
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  2. #2
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    Quote Originally Posted by JaysFan31 View Post
    Let R be any commutative ring and let a1,......,an be elements of R. We define
    (a1,.....,an) to be the set {r1a1+...rnan for all i, ri in R}.
    Prove that (a1,......,an) is an ideal of R. We call it the ideal generated by a1,......, an.
    Let us show that,
    S=\{r_1a_1+...+r_na_n\}
    Has property that,
    xS\subseteq S, Sx\subseteq S
    Well that is simple to show,
    Any element of xS can be expressed as,
    \{xr_1a_1+...+xr_na_n\}
    And every xr_i\in R (for it is closed since it is a ring).

    Now the other way around Sx\subseteq S is also true for any element of Sx can be expressed as,
    r_1a_1x+...+r_na_nx
    But since R is a commutative ring,
    xr_1a_1+...+xr_na_n.
    That part is finished.

    The question remains to show that this generating set is an additive subgroup of the ring. I am not going to show that (in fact it is simple to) but you show be familar from group theory that these generating sets always forms a subgroup (if one element only then it is a cyclic subgroup).
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