on dimension of ab5∗ modules

‎in this article‎, ‎the $alpha$-shortness of $r$-modules in the category of $ab5^*$ were studied‎. ‎we‎ ‎proved that if $m$ is an $r$-module in $ab5^*$ which has noetherian dimension‎ ‎and ${n_i}$ is a family of submodules of $m$ such that $dim,frac{m}{n_i} leq alpha$‎, ‎for each $i$‎, ‎then $dim,frac{m}{cap n_i} leq alpha$ (this is the duall of a famous result due to lenegan)‎. ‎using this‎, ‎we give an structure theorem for $alpha$-short modules in $ab5^*$ and finally‎, ‎we classify $alpha$-short modules in this category‎.