6.6 Rings and fields 6.6.1 Rings 6 Definition 21: A ring is an Abelian group r,+ with an additional associative binary operation (denoted such that for all a, b, CER, (1)a·(b+c)=a·b+a·c, (2)(b+c)·a=b·a+c·a We write oER for the identity element of the group R,+ t Fora eR. we write -a for the additive inverse of a o Remark: Observe that the addition operation is always commutative while the multiplication need not be e Observe that there need not be inverses for multiplication.6.6 Rings and fields 6.6.1 Rings  Definition 21: A ring is an Abelian group [R, +] with an additional associative binary operation (denoted ·) such that for all a, b, cR,  (1) a · (b + c) = a · b + a · c,  (2) (b + c) · a = b · a + c · a.  We write 0R for the identity element of the group [R, +].  For a R, we write -a for the additive inverse of a.  Remark: Observe that the addition operation is always commutative while the multiplication need not be.  Observe that there need not be inverses for multiplication