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Let $p$ be a prime number. In this paper, we study cyclic codes over the ring$ \Z_p[u, v]/\langle u^2, v^2, uv-vu\rangle$. We find a unique set ofgenerators for these codes. We also study the rank and the Hamming distance ofthese codes. We obtain all except one ternary optimal code of length 12 as theGray image of the cyclic codes over the ring $ \Z_p[u, v]/\langle u^2, v^2,uv-vu\rangle$. We also characterize the $p$-ary image of these cyclic codesunder the Gray map.

Cyclic codes over the ring $ \Z_p[u, v]/\langle u^2, v^2, uv-vu\rangle$