id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejpam-5224	Laaraj, Mounir; Abdelalim, Seddik; Elmouki, Ilias	One Resolution of the Conjecture between the Projective Dimension of a Simple Module S of an Artinian Ring on Zero Radical’s Cube and the First Bifunctor Extension on S	2024	14	.pdf	application/pdf	5095	285	73	Here are the calculations of the projective resolutions of simple modules (S(i))1≤i≤5, 0 −→ P (3)⊕ P (4) −→ P (2)2 −→ P (1) −→ S(1) −→ 0 REFERENCES 1867 0 −→ P (3)⊕ P (4) −→ P (2) −→ S(2) −→ 0 0 −→ P (5) −→ P (3) −→ S(3) −→ 0 0 −→ P (5) −→ P (4) −→ S(4) −→ 0 0 −→ P (5) −→ P (4) −→ S(5) −→ 0 = 0 because S(1) is an injective module and we have, 0 −→ P (5) −→ P (4) −→ P (2) −→ P (1) −→ S(1) −→ 0 which implies that pd(S) = 3 7.2.	cache/ejpam-5224.pdf	txt/ejpam-5224.txt