A GG NOT FH SEMISTAR OPERATION ON MONOIDS

Year 2017, Volume: 22 Issue: 22, 39 - 44, 11.07.2017

Ryuki Matsuda

https://doi.org/10.24330/ieja.325920

Cited By: 2

Abstract

Let  $S$  be a  g-monoid with
quotient group  q$(S)$. Let  $\bar {\rm F}(S)$ (resp., F$(S)$,
f$(S)$)  be the  $S$-submodules of  q$(S)$ (resp., the  fractional
ideals of  $S$,  the finitely generated  fractional ideals of
$S$). Briefly, set  f := f$(S)$, g := F$(S)$, h := $\bar{\rm
F}(S)$, and let   $\{\rm{x,y}\}$  be a subset of the set  $\{$f,
g, h$\}$  of symbols. For a semistar operation  $\star$ on  $S$,
if  $(E + E_1)^\star = (E + E_2)^\star$ implies  ${E_1}^\star =
{E_2}^\star$  for every  $E \in$  x  and every  $E_1, E_2 \in$ y,
then  $\star$  is called  xy-cancellative. In this paper, we
prove that  a  gg-cancellative semistar operation
need not be  fh-cancellative.

Keywords

Semistar operation, monoid

References

• M. Fontana and K. A. Loper, Kronecker function rings: a general approach, Ideal theoretic methods in commutative algebra (Columbia, MO, 1999), Lec- ture Notes in Pure and Appl. Math., 220 (2001), 189-205.
• M. Fontana and K. A. Loper, Cancellation properties in ideal systems: A classification of e.a.b. semistar operations, J. Pure Appl. Algebra, 213(11) (2009), 2095-2103.
• M. Fontana, K. A. Loper and R. Matsuda, Cancellation properties in ideal systems: an e.a.b. not a.b. star operation, Arab. J. Sci. Eng. ASJE. Math., 35 (2010), 45-49.
• R. Gilmer, Multiplicative Ideal Theory, Pure and Applied Mathematics, 12, Marcel Dekker, Inc., New York, 1972.
• R. Gilmer, Commutative Semigroup Rings, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1984.
• F. Halter-Koch, Ideal Systems: An Introduction to Multiplicative Ideal The- ory, Monographs and Textbooks in Pure and Applied Mathematics, 211, Mar- cel Dekker, Inc., New York, 1998.
• R. Matsuda, Multiplicative Ideal Theory for Semigroups, 2nd ed., Kaisei, Tokyo, 2002.
• R. Matsuda, Note on g-monoids, Math. J. Ibaraki Univ., 42 (2010), 17-41.
• R. Matsuda, Cancellation properties in ideal systems of monoids, Int. Electron. J. Algebra, 9 (2011), 61-68.
• R. Matsuda, Note on cancellation properties in ideal systems, Comm. Algebra, 43(1) (2015), 23-28.
• R. Matsuda, A gg not gh semistar operation on monoids, Bull. Allahabad Math. Soc., 31(1) (2016), 111-119.
• D. G. Northcott, Lessons on Rings, Modules and Multiplicities, Cambridge Univ. Press, London, 1968.