Commutative ring
In algebra, commutative ring is a set of elements in which you can add and multiply and have multiplication distribute over addition. An example of a commutative ring is the set of integers. If we add two integers, we get an integer and if we multiply two integers we get another integer. Moreover, multiplication distributes in the sense that if a, b, and c are integers, then c*(a+b)=ca+cb.
Commutative Ring Media
Spec (Z) contains a point for the zero ideal. The closure of this point is the entire space. The remaining points are the ones corresponding to ideals (p), where p is a prime number. These points are closed.
The universal property of S ⊗R T* states that for any two maps S → W and T → W which make the outer quadrangle commute, there is a unique map S ⊗R T → W which makes the entire diagram commute.
The cubic plane curve (red) defined by the equation y2 = x2(x + 1) is singular at the origin, i.e., the ring k[x, y] / y2 − x2(x + 1), is not a regular ring. The tangent cone (blue) is a union of two lines, which also reflects the singularity.
The twisted cubic (green) is a set-theoretic complete intersection, but not a complete intersection.
A pair of pants is a cobordism between a circle and two disjoint circles. Cobordism classes, with the cartesian product as multiplication and disjoint union as the sum, form the cobordism ring.
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