On some annihilator conditions over commutative rings

It is proved that if a commutative ring R has finite Goldie dimension then T(R), the total ring of quotient of R, is semilocal, It is also shown that a ring R is CS if and only if AnnI + AnnJ = R for any ideals I and J such that I boolean AND J = 0 (H). As a consequence, we characterize completely CS rings. A ring is pseudo-PIF if it satisfies Property (H) for any principal ideals I and J. Several examples are given and completely pseudo-PIF ring's arc characterized as arithmetical rings. Finally, we prove that if R is pseudo-PIF with acc on annihilators, then T(R) is quasi-Frobenius, extending known results on the subject.