On Unital Covers of Commutative Rings of Small Orders by Proper Subrings

A cover of a finite ring R is a set of proper subrings of R whose union is R. If R has a multiplicative identity \(1_R\) $1_R$ and every member of a cover of R contains \(1_R\) $1_R$ , then such cover is called unital. In this paper, we determine all commutative rings R having a unital cover with \(|R|=p_1^{\alpha _1}p_2^{\alpha _2}\cdots p_k^{\alpha _k}\) $|R|=p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}\cdots p_k^{{\alpha }_k}$ where \(p_1,\cdots , p_k\) $p_1,\cdots ,p_k$ are distinct primes and \(1\le \alpha _i\le 4\) $1\le {\alpha }_i\le 4$ for each i.