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A GG NOT FH SEMISTAR OPERATION ON MONOIDS

Year 2017, Volume: 22 Issue: 22, 39 - 44, 11.07.2017
https://doi.org/10.24330/ieja.325920

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.

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.
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  • 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.
Year 2017, Volume: 22 Issue: 22, 39 - 44, 11.07.2017
https://doi.org/10.24330/ieja.325920

Abstract

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.
There are 12 citations in total.

Details

Subjects Mathematical Sciences
Journal Section Articles
Authors

Ryuki Matsuda This is me

Publication Date July 11, 2017
Published in Issue Year 2017 Volume: 22 Issue: 22

Cite

APA Matsuda, R. (2017). A GG NOT FH SEMISTAR OPERATION ON MONOIDS. International Electronic Journal of Algebra, 22(22), 39-44. https://doi.org/10.24330/ieja.325920
AMA Matsuda R. A GG NOT FH SEMISTAR OPERATION ON MONOIDS. IEJA. July 2017;22(22):39-44. doi:10.24330/ieja.325920
Chicago Matsuda, Ryuki. “A GG NOT FH SEMISTAR OPERATION ON MONOIDS”. International Electronic Journal of Algebra 22, no. 22 (July 2017): 39-44. https://doi.org/10.24330/ieja.325920.
EndNote Matsuda R (July 1, 2017) A GG NOT FH SEMISTAR OPERATION ON MONOIDS. International Electronic Journal of Algebra 22 22 39–44.
IEEE R. Matsuda, “A GG NOT FH SEMISTAR OPERATION ON MONOIDS”, IEJA, vol. 22, no. 22, pp. 39–44, 2017, doi: 10.24330/ieja.325920.
ISNAD Matsuda, Ryuki. “A GG NOT FH SEMISTAR OPERATION ON MONOIDS”. International Electronic Journal of Algebra 22/22 (July 2017), 39-44. https://doi.org/10.24330/ieja.325920.
JAMA Matsuda R. A GG NOT FH SEMISTAR OPERATION ON MONOIDS. IEJA. 2017;22:39–44.
MLA Matsuda, Ryuki. “A GG NOT FH SEMISTAR OPERATION ON MONOIDS”. International Electronic Journal of Algebra, vol. 22, no. 22, 2017, pp. 39-44, doi:10.24330/ieja.325920.
Vancouver Matsuda R. A GG NOT FH SEMISTAR OPERATION ON MONOIDS. IEJA. 2017;22(22):39-44.