Question

Verify that$\mathit{I}\mathbf{=}\left\{0,3,6,9,12\right\}$ is an ideal in${\mathbf{\mathbb{Z}}}_{\mathbf{15}}$ and list all its distinct cosets.

Short answer

The list of distinct cosets is$\left\{0,3,6,9,12\right\};\left\{1,4,5,10,13\right\};\left\{2,5,8,11,14\right\}$.

Step-by-step solution

Theorem statement

A set $I=\left\{rc/r\in R\right\}$ is said to be ideal if R is a commutative ring with identity, $c\in R$ and $I$is the set of all multiples of c inR.

Proof part

The given set is ${\mathbf{\mathbb{Z}}}_{\mathbf{15}}$, which is a commutative ring with identity.

And the one more set, $I=\left\{0,3,6,9,12\right\}$ is the set of multiples of 3, so by the theorem stated in step 1, $I$is an ideal. So, the cosets will be:

$\left\{0,3,6,9,12\right\};\left\{1,4,5,10,13\right\};\left\{2,5,8,11,14\right\}.$

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