A class of Artinian local rings of homogeneous type

Let $I$ be an ideal in a regular local ring $(R,n)$, we will find bounds on the first and the last Betti numbers of $(A,m)=(R/I,n/I)$. if $A$ is an Artinian ring of the embedding codimension $h$, $I$ has the initial degree $t$ and $mu(m^t)=1$, we call $A$ a {it $t-$extended stretched local ring}. This class of local rings is a natural generalization of the class of stretched local rings studied by Sally, Elias and Valla. For a $t-$extended stretched local ring, we show that ${h+t-2choose t-1}-h+1leq tau(A)leq {h+t-2choose t-1}$ and $ {h+t-1choose t}-1 leq mu(I) leq {h+t-1choose t}$. Moreover $tau(A)$ reaches the upper bound if and only if $mu(I)$ is the maximum value. Using these results, we show when $beta_i(A)=beta_i(gr_m(A))$ for each $igeq 0$. Beside, we will investigate the rigid behavior of the Betti numbers of $A$ in the case that $I$ has initial degree $t$ and $mu(m^t)=2$. This class is a natural generalization of {it almost stretched local rings} again studied by Elias and Valla. Our research extends several results of two papers by Rossi, Elias and Valla.