On image summand coinvariant modules and kernel summand invariant modules


KESKİN TÜTÜNCÜ D. , Kuratomi Y., Shibata Y.

TURKISH JOURNAL OF MATHEMATICS, cilt.43, sa.3, ss.1456-1473, 2019 (SCI İndekslerine Giren Dergi) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 43 Konu: 3
  • Basım Tarihi: 2019
  • Doi Numarası: 10.3906/mat-1808-40
  • Dergi Adı: TURKISH JOURNAL OF MATHEMATICS
  • Sayfa Sayıları: ss.1456-1473

Özet

In this paper we introduce the concept of im-summand coinvariance and im-small coinvariance; that is, a module M over a right perfect ring is said to be im-summand (im-small) coinvariant if, for any endomorphism phi of P such that Im phi is a direct summand (a small submodule) of P , phi(ker v) subset of ker v, where (P, v) is the projective cover of M. We first give some fundamental properties of im-summand coinvariant modules and im-small coinvariant modules, and we prove that, for modules M and N over a right perfect ring such that N is a small epimorphic image of M, M is N-im-summand coinvariant if and only if M is (im-coclosed) N-projective. Moreover, we introduce ker-summand invariance and ker-essential invariance as the dual concept of im-summand coinvariance and im-small coinvariance, respectively, and show that, for modules M and N such that N is isomorphic to an essential submodule of M, M is N-ker-summand invariant if and only if M is (ker-closed) N-injective.