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Let $\mathbb{F}_{2^m}$ be a finite field of $2^m$ elements, and$R=\mathbb{F}_{2^m}[u]/\langleu^k\rangle=\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}+\ldots+u^{k-1}\mathbb{F}_{2^m}$($u^k=0$) where $k$ is an integer satisfying $k\geq 2$. For any odd positiveinteger $n$, an explicit representation for every self-dual cyclic code over$R$ of length $2n$ and a mass formula to count the number of these codes aregiven first. Then a generator matrix is provided for the self-dual and$2$-quasi-cyclic code of length $4n$ over $\mathbb{F}_{2^m}$ derived by everyself-dual cyclic code of length $2n$ over $\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$and a Gray map from $\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ onto$\mathbb{F}_{2^m}^2$. Finally, the hull of each cyclic code with length $2n$over $\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ is determined and all distinctself-orthogonal cyclic codes of length $2n$ over$\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ are listed.

On self-duality and hulls of cyclic codes over $\frac{\mathbb{F}_{2^m}[u]}{\langle u^k\rangle}$ with oddly even length