A Review of Commutative Ring Theory Mathematics Undergraduate Seminar: Toric Varieties

Definition 1.1 (Rings). The algebraic structure “ring” R is a set with two binary operations + and ·, respectively named addition and multiplication, satisfying • (R,+) is an abelian group (i.e. a group with commutative addition), • is associative (i.e. 8a, b, c 2 R, (a · b) · c = a · (b · c)) , • and the distributive law holds (i.e. 8a, b, c 2 R, (a+ b) · c = a · c+ b · c, a · (b+ c) = a · b+ a · c.) Moreover, the ring is commutative if multiplication is commutative. The ring has an identity, conventionally denoted 1, if there exists an element 1 2 R s.t. 8a 2 R, 1 · a = a · 1 = a. From now on, all rings considered will be commutative rings (after all, this is a review of commutative ring theory...) Since we will be talking substantially about the complex field C, let us recall the definition of such structure.