On the complete intersection conjecture of Murthy

Suppose A=k[X1,X2,…,Xn]A=k[X1,X2,…,Xn] is a polynomial ring over a field k and I is an ideal in A  . M.P. Murthy conjectured that μ(I)=μ(I/I2)μ(I)=μ(I/I2), where μ denotes the minimal number of generators. Recently, Fasel [3] settled this conjecture, affirmatively, when k   is an infinite perfect field, with 1/2∈k1/2∈k (always). We are able to do the same, when k is an infinite field. In fact, we prove similar results for ideals I   in a polynomial ring A=R[X]A=R[X], that contains a monic polynomial and R is essentially smooth algebra over an infinite field k, or R is a regular ring over a perfect field k.