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Commutative Algebra

Field: Ring & Field Theory

The study of commutative rings and their ideals, and modules over such rings.

Sequence of Expressions

Let RR be a set equipped with two binary operations, addition (++) and multiplication (⋅\cdot). RR is a commutative ring if (R,+,⋅)(R, +, \cdot) satisfies the following axioms for all a,b,c∈Ra, b, c \in R: \n\n1. (R,+)(R, +) is an abelian group: \n a. Associativity: (a+b)+c=a+(b+c)(a+b)+c = a+(b+c) \n b. Commutativity: a+b=b+aa+b = b+a \n c. Identity: There exists an additive identity 0∈R0 \in R such that a+0=aa+0 = a. \n d. Inverse: For every a∈Ra \in R, there exists an additive inverse −a∈R-a \in R such that a+(−a)=0a+(-a) = 0. \n\n2. (R,⋅)(R, \cdot) is a commutative monoid: \n a. Associativity: (a⋅b)⋅c=a⋅(b⋅c)(a \cdot b) \cdot c = a \cdot (b \cdot c) \n b. Commutativity: a⋅b=b⋅aa \cdot b = b \cdot a \n c. Identity: There exists a multiplicative identity 1∈R1 \in R such that a⋅1=aa \cdot 1 = a. \n (Note: We typically require 0≠10 \neq 1 for non-trivial rings.)\n\n3. Distributivity: Multiplication distributes over addition: a⋅(b+c)=(a⋅b)+(a⋅c)a \cdot (b+c) = (a \cdot b) + (a \cdot c).