  In this note, we discuss the consistency conditions which a discrete $Z_N$
symmetry should satisfy in order that it is not violated by gauge and
gravitational instantons. As examples, we enlist all the $Z_N$ ${\cal
R}$-symmetries as well as non-${\cal R}$ $Z_N$ symmetries (N=2,3,4) in the
minimally supersymmetric standard model (MSSM) that are free from gauge and
gravitational anomalies. We show that there exists non-anomalous discrete
symmetries that forbid Baryon number violation up to dimension 6 level (in
superspace). We also observe that there exists no non-anomalous $Z_3$ ${\cal
R}$-symmetry in the MSSM. Furthermore, we point out that in a theory with one
Majorana spin 3/2 gravitino, a large class of $Z_4$ ${\cal R}$-symmetries are
violated in the presence of Eguchi-Hanson (EH) gravitational instanton. This is
also in general true for higher $Z_N$ ${\cal R}$-symmetries. We also notice
that in 4 dimensional ${\cal N}=1$ supergravity, the global $U(1)$ ${\cal
R}$-symmetry is always violated by the EH instanton irrespective of the matter
content of the theory.
