Let I be an ideal of a ring R and [R:I]={x∈R ∣ rx∈I for all r∈R}
(i) Let a∈[R:I] and b∈[R:I]. Then ra∈I and rb∈I for all r∈R, and therefore, r(a−b)=ra−rb∈I for all r∈R. We conclude that a−b∈[R:I], and thus [R:I] is a subgroup of the additive group of a ring R.
Let a∈[R:I] and s∈R be arbitrary. It follows that ra∈I for all r∈R. Since I is an ideal, s(ra)∈I for all s∈R and r(as)=(ra)s∈I for all r∈R. Consequently, ra∈[R:I] and as∈[R:I] for all s,r∈R, and [R:I] is an ideal of a ring R.
(ii) If a∈I, then ra∈I for all r∈R. It follows that a∈[R:I], and thus I is a subset of [R:I].
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