Show that the ring (Z4, +4, .4) is an integral domain.

Question

Show that the ring (Z4, +4, .4) is an integral domain.

Suresh Chand · Accepted Answer

Solution: Z4 = {0, 1, 2, 3} +4 0 1 2 3 0 0 1 2 3 1 1 2 3 0 2 2 3 0 1 3 3 0 1 2 .4 0 1 2 3 0 0 0 0 0 1 0 1 2 3 2 0 2 0 2 3 0 3 2 1 R1: (Z4, +) is an abelian group Here, (Z4, +) closure according to table under + It is associative Identity element is 0 Inverse of 0 is 0 , 1 is 3, 2 is 2, 3 is 1 It is also sommutative in 0 +4 1 = 1 = 1 +4 0 R2: Associative 0 ×4 (1 ×4 2) = 0 = (0 ×4 1) ×4 2 R3: Distributive 1 ×4 (2 +4 3) = 1 ×4 1 = 1 (1 +4 2) ×4 3 = 3 ×4 3 = 1 So, it is a ring Now, 0 ×4 1 = 1 ×4 0 = 0 So, it is commutative ring Aslo, multiplicative identity 1 ∈ Z4 so, it is commutative ring with identity Also, z4 = {0, 1, 2, 3} For a ring to be an integral domain, it should not have a zero divisor. So the given ring is not an integral domain.

Show that the ring (Z₄, +₄, .₄) is an integral domain.

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Show that the ring (Z4, +4, .4) is an integral domain.

Leave your Answer:

Show that the ring (Z₄, +₄, .₄) is an integral domain.