Ring • Definition ○ A ring is a set R equipped with two operations § +:R×R→R § ⋅:R×R→R ○ such that § (R,+) is an abelian group § ⋅ is associative § ∃1∈R s.t. 1⋅r=r=r⋅1 § Distributive property: □ ∀a,b,c∈R □ a⋅(b+c)=a⋅b+a⋅c □ (a+b)⋅c=a⋅c+b⋅c • Notes ○ 1 is called the multiplicative identity ○ Dummit-Foote don t require the multiplicative identity ○ ⋅ is not necessarily commutative ○ R is not a group under ⋅, because inverses may not exist ○ We will typically denote multiplication of r,s∈R by rs ○ Typically 1 will denote the multiplicative identity ○ And 0 will denote the identity of (R,+)
