Define ring with an example. Compute the product in the given ring (-3, 5) (2, -4) in Z4 x Z11.

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A non-empty set R together with two binary operator + and denoted by <R, +, > is called ring if the following conditions are statisfied

R1: <R, +> is an abelian group

R2: closed: ∀ a, b, ∈ R, ab ∈ R

R3: Associative: ∀ a,b, c ∈ R    ,    (ab)c = a(bc)

R4: Distributive: ∀ a,b, c ∈ R

a (b + c) = ab + bc           (left)

(a + b) c = ac + bc          (right)

Ex: The set of all integer Z is a ring under binary operation addition and multiplication.

Problem Part:

Given ring is Z4 x Z11

Given, (-3, 5), (2, -4)

Product is given as

(-3, 5) (2, -4) = (-6, -20)

Divide (-6) by 4 we get remainder -2

so, -2 + 4 = 2

Divide (-20) by 11 we get remainder -9

so, -9 + 11 = 2

Therefore The product (-3, 5) and (2, -4) is (2, 2) in ring Z4 x Z11

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