The Extension Dimension of Abelian Categories

Let $\text{A}$ be an abelian category having enough projective objects and enough injective objects. We prove that if $\text{A}$ admits an additive generating object, then the extension dimension and the weak resolution dimension of $\text{A}$ are identical, and they are at most the representation dimension of $\text{A}$ minus two. By using it, for a right Morita ring $\text{La}$, we establish the relation between the extension dimension of the category $\mathrm{m}\mathrm{o}\mathrm{d}\text{La}$ of finitely generated right $\mathrm{\Lambda }$-modules and the representation dimension as well as the right global dimension of $\mathrm{\Lambda }$. In particular, we give an upper bound for the extension dimension of $\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{\Lambda }$ in terms of the projective dimension of certain class of simple right $\mathrm{\Lambda }$-modules and the radical layer length of $\mathrm{\Lambda }$. In addition, we investigate the behavior of the extension dimension under some ring extensions and recollements.