Define ring. Show that set of positive integers with respect to addition and multiplication operation is not a ring.

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Samikshya B. Chhetri · Accepted Answer

To show that the set of positive integers with respect to addition and multiplication operation is not a ring, we need to show that it fails to satisfy at least one of the axioms of a ring. (a + b) + c = a + (b + c) (associativity of addition) a + b = b + a (commutativity of addition) There exists an element 0 in R such that a + 0 = a for all a in R (existence of additive identity) For each a in R, there exists an element -a in R such that a + (-a) = 0 (existence of additive inverse) (a × b) × c = a × (b × c) (associativity of multiplication) a × (b + c) = a × b + a × c (distributivity of multiplication over addition) (a + b) × c = a × c + b × c (distributivity of addition over multiplication) There exists an element 1 in R such that a × 1 = a for all a in R (existence of multiplicative identity) Let R be the set of positive integers with respect to addition and multiplication operation. First, we observe that R satisfies axioms 1-5, and axiom 8. However, R fails to satisfy axiom 6. In particular, there exist positive integers a, b, and c such that: a × (b + c) ≠ a × b + a × c For example, let a = 2, b = 3, and c = 4. Then: a × (b + c) = 2 × (3 + 4) = 2 × 7 = 14 and a × b + a × c = 2 × 3 + 2 × 4 = 6 + 8 = 14 Since a × (b + c) ≠ a × b + a × c, we see that R does not satisfy axiom 6, and therefore, it is not a ring.

Define ring. Show that set of positive integers with respect to addition and multiplication operation is not a ring.

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Define ring. Show that set of positive integers with respect to addition and multiplication operation is not a ring.

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