Explain why the natural numbers are not a ring. THEOREM 8.2.6 Zn IS A RING Let n

Question

Explain why the natural numbers are not a ring.

THEOREM 8.2.6 Zn IS A RING Let n E N. Then Z, satisfies the following arithmetical properties Properties of Addition Associativity. For every a, b, and in Z (aa(b+2) Commutativity. For every a and b in Z,aa. . Identity. The element 0 in Z, is an additive identity element For every-in-a + 0 = . Additive Inverses. For every a in Z, there exists an additive inverse, -a in Z. such that + (--) = 0. Properties of Multiplication A Z,,G--).-=-. (hi). ssociativity. For every a, b, and c in Commutativity. For every a and b in Z, ab-b.a. Identity. The element 1 in Z, is a multiplicative identity element: For every a in Z, a 1 a. . Properties Relating Addition and Multiplication Distributivity. For everya, b, and c in Zn, a . (D+ Z) = ( . b) + (·Z). .

Explanation / Answer

The Natural numbers N, do not even possess additive inverses so they are neither a field nor a ring.

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