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Commutative ringsCommutative rings|
I is an ideal if a,b ∈ I ⇒ a + b ∈ I and ∀r ∈ R { r·I ⊆ I, I·r ⊆ I</ms:remark> |
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I is an ideal if a,b ∈ I ⇒ a + b ∈ I and ∀r ∈ R (r·I ⊆ I)</ms:remark> |
A commutative ring is a ring R = (R, + ,−,0,·) such that · is commutative: x·y = y ·x.
Remark: Idl(R) = { all ideals of R}
I is an ideal if a,b ∈ I ⇒ a + b ∈ I
and ∀r ∈ R { r·I ⊆ I, I·r ⊆ I
Let R and S be commutative rings with identity. A morphism from R to S is a function h : R→S that is a homomorphism: h(x + y) = h(x) + h(y) and h(x·y) = h(x)·h(y).
Remark: It follows that h(0) = 0 and h(−x) = −h(x).
0 is a zero for ·: 0·x = x and x·0 = 0.
(Z, + ,−,0,·), the ring of integers with addition, subtraction, zero, and multiplication.