AMC 125: Rings and Modular Arithmetic

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TOPIC: RINGS AND MODULAR ARITHMETIC

(Grimaldi, 673 - 679)

Definition 1.0 Ring Structure

Let R be a nonempty set on which we have two closed binary operations, denoted by + and ⋅ (which may be quite different from the ordinary addition and multiplication to which we are accustomed). Then (R, +, ⋅) is a ring if for all a,b,c ∈R,the following conditions are satisfied:

a) a + b = b + a (Commutative Law of + )

b) a + (b + c) = (a +b) + c (Associative Law of +)

c) There exists z ∈R such that a+z = z+a=a foe every a∈R.(Existence of an identity for +)

d) For each a ∈R there is an element b∈R with a + b = b + a = z (Existence of inverse under +)

e) a ⋅(b⋅c)=(a⋅b)⋅c (Associative Law of ⋅)

f) a⋅(b+c)=a⋅b+a⋅c (b + c) ⋅a=b⋅a+c⋅a ( Distributive Laws of . over +)

Note: + refers to ring addition and ⋅ refers to ring multiplication

Definition 2.0 Ring with Unity

Let (R, +, ⋅) be a ring.

a) If ab=ba for all a,b ∈R, then R is called a commutative ring.

b) The ring R is said to have no proper divisors of zero if for all a∈R, b∈R,ab=z⇒a=z or b=z.

c) If an element u∈R is such that u≠z and au=ua=a for all a∈R,we call u a unity, or multiplicative identifty , of R. Here R is called a ring with unity.



Graded Activity: Study Example 14.3, page 675 (Grimaldi). Answer the following question (to be submitted): Define the binary operations ⊕ and ⊙ on Z by x⊕y=x+y-7 and x⊙y=x+y-3xy, for all x,y∈Z. Question: Explain (completely) why (Z,⊕,⊙) is not a ring.

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