  In this paper we introduce the notion of $\lambda$-constacyclic codes over
finite rings $R$ for arbitary element $\lambda$ of $R$. We study the
non-invertible-element constacyclic codes (NIE-constacyclic codes) over finite
principal ideal rings (PIRs). We determine the algebraic structures of all
NIE-constacyclic codes over finite chain rings, give the unique form of the
sets of the defining polynomials and obtain their minimum Hamming distances. A
general form of the duals of NIE-constacyclic codes over finite chain rings is
also provided. In particular, we give a necessary and sufficient condition for
the dual of an NIE-constacyclic code to be an NIE-constacyclic code. Using the
Chinese Remainder Theorem, we study the NIE-constacyclic codes over finite
PIRs. Furthermore, we construct some optimal NIE-constacyclic codes over finite
PIRs in the sense that they achieve the maximum possible minimum Hamming
distances for some given lengths and cardinalities.
