extended ideals in residuated lattices

In this paper, we generalized the notion of extended ideals and stable ideals associated to a subset b of a residuated  lattices l and we discuss what kind of residuated lattices have extended ideals. then we investigate the related  properties  of them. we show that if  l  is an involutive residuated lattice, then ei(b) is a stable ideal relative to b  and so, ei(b)= $bigcap $ { j : j  is stable relative to b  and $i subseteq j $}. we also give a characterization of this extended ideal: ei(b)= (b] --> i  in the complete heyting algebra $(operatorname{id}(l), wedge, vee, rightarrow,{0}, l) $. we also prove that the class s(b)  of all stable ideals relative to $b subseteq l $ is a complete heyting algebra. finally, we prove  that the set of extended ideals and the set of extended filters on involutive residuated lattices are one-to-one  correspondence.