  We study the structure of the Eisenstein component of Hida's ordinary p-adic
Hecke algebra attached to modular forms, in connection with the companion forms
in the space of modular forms (mod p). We show that such an algebra is a
Gorenstein ring if certain space of modular forms (mod p) having companions is
one-dimensional; and also give a numerical criterion for this
one-dimensionality. This in part overlaps with a work of Skinner and Wiles; but
our method, based on a work of Ulmer, is totally different. We then consider
consequences of the above mentioned Gorenstein property. We especially discuss
the connection with the Iwasawa theory.
