  We study the conjecture that if $I,J$ are $\mu-$primary in a regular local
ring $(R,\mu)$ with dim$(R)=n$, then $\frac{I \cap J}{IJ} \cong \Tor_1(R/I,
R/J)$ needs at least $n$ generators, and a related conjecture about the number
of generators of the first Koszul homology module of an Artinian local ring
$(A,m)$. In this manuscript, we focus our attention on the complete
intersection defect of the Artinian ring and its quotient by the Koszul
elements. We prove that the number of generators of the first Koszul homology
module of $x_1,...,x_n \in m$ on an Artinian local ring $(A,m)$ is at least $n+
\cid(A)- \cid(\frac{A}{(x_1,...,x_n)A})$, where $\cid A$ denotes the complete
intersection defect of the Artinian local ring $A$.
