Idempotent elements of Z_12
Moderate Ring Theory question for IIT JAM · CUET-PG. Choose an option, check your answer, then read the detailed solution.
Question RT-002
Let $R=\mathbb Z_{12}$ with addition and multiplication modulo $12$. An element $x\in R$ is called idempotent if $x^2=x$. How many idempotent elements does $R$ contain?
Detailed Solution
Checking $x^2\bmod12$ against $x$ for each residue: $0^2=0$, $1^2=1$, $4^2=16\equiv4$, and $9^2=81\equiv9\pmod{12}$ all satisfy $x^2=x$. Direct computation shows none of the remaining residues ($2,3,5,6,7,8,10,11$) satisfy the condition. So the complete set of idempotents is $\{0,1,4,9\}$, exactly $4$ elements. Thus the correct option is $\boxed{\mathrm{C}}$.Concepts tested: Idempotent elementsModular arithmetic
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