Answer to Question #236513 in Abstract Algebra for 123

Question

Consider { 0,2,4} as a subset of Z₆ . show it is a subring. does it have unity?

Answer 1

Let us consider $R=\{ 0,2,4\}$ as a subset of $\Z_6$. Let us show that $R$ is a subring of $\Z_6.$ Since $0-0=0\in\Z_6,\ 2-0=2\in\Z_6,\ 4-0=4\in\Z_6,\ 0-2=4\in\Z_6,\ 2-2=0\in\Z_6,\ 4-2=2\in\Z_6,\ 0-4=2\in\Z_6,\ 2-4=4\in\Z_6,\ 4-4=0\in\Z_6,\$

we conclude that the operation of substraction is closed on $R.$

Taking into account that $0\cdot 0=0\cdot 2=2\cdot 0=4\cdot 0=0\cdot 4=0\in\Z_6,\ 2\cdot 2=4\in\Z_6, \ 2\cdot 4=4\cdot 2=2\in\Z_6$ and $4\cdot 4=4\in\Z_6,$ we conclude that the operation of multiplication is closed on $R.$

Therefore, $R$ is a subring of $\Z_6.$

Taking into account that $4\cdot 4=4, 4\cdot 0=0\cdot 4=0,4\cdot 2=2\cdot 4=2,$ we conclude that $4$ is unity of $R.$