A right ternary near-ring (RTNR) is an algebraic system which is a group under binary addition and a ternary semigroup under ternary product satisfying the right distributive law. A 3-uniform hypergraph is a hypergraph in which all of its hyperedges are of size 3. In this paper, a 3-uniform hypergraph of an RTNR ‘N’, denoted by  is defined. In a zero-symmetric RTNR, some of the properties of  are proved. As a special case,  is studied in depth. The3-uniform friendship hypergraphs are associated to RTNR    in which the ternary product of three distinct elements is zero. The hyperedges of these 3-uniform friendship hypergraphs are treated as blocks and it is shown that the 3-uniform friendship hypergraphs of    are balanced incomplete block designs (BIBDs).

RTNR, 3-uniform hypergraph, 3-uniform friendship hypergraph, BIBD, zero-symmetric RTNR.