Revisiting Baer elements

Year 2025, Volume: 37 Issue: 37, 201 - 219, 14.01.2025

Amartya Goswami , Themba Dube

https://doi.org/10.24330/ieja.1521082

Abstract

The objective of this paper is to extend certain properties observed in $d$-ideals of rings and $d$-elements of frames to Baer elements in multiplicative lattices. Additionally, we present results concerning these elements that have not been addressed in the study of $d$-ideals of rings. Furthermore, we introduce Baer closures and explore Baer maximal, prime, semiprime and meet-irreducible elements.

Keywords

Multiplicative lattice, compact element, annihilator element, Baer element, Baer closure

References

• F. Alarcon, D. D. Anderson and C. Jayaram, Some results on abstract commutative ideal theory, Period. Math. Hungar., 30(1) (1995), 1-26.
• D. D. Anderson, Multiplicative Lattices, Ph.D dissertation, The University of Chicago, 1974.
• D. D. Anderson, Abstract commutative ideal theory without chain condition, Algebra Universalis, 6(2) (1976), 131-145.
• D. D. Anderson, C. Jayaram and P. A. Phiri, Baer lattices, Acta Sci. Math. (Szeged), 59(1-2) (1994), 61-74.
• F. Azarpanah, O. A. S. Karamzadeh and A. R. Aliabad, On $z^0$-ideals in $C(X)$, Fund. Math., 160(1) (1999), 15-25.
• F. Azarpanah, O. A. S. Karamzadeh and A. R. Aliabad, On ideals consisting entirely of zero divisors, Comm. Algebra, 28(2) (2000), 1061-1073.
• B. Banaschewski, On certain localic nuclei, Cahiers Topologie Geom. Differentielle Categ., 35(3) (1994), 227-237.
• B. Banaschewski and R. Harting, Lattice aspects of radical ideals and choice principles, Proc. London Math. Soc., 50(3) (1985), 385-404.
• P. Bhattacharjee, Maximal $d$-elements of an algebraic frame, Order, 36(2) (2019), 377-390.
• M. Contessa, A note on Baer rings, J. Algebra, 118(1) (1988), 20-32.
• R. P. Dilworth, Abstract residuation over lattices, Bull. Amer. Math. Soc., 44(4) (1938), 262-268.
• R. P. Dilworth, Abstract commutative ideal theory, Pacific J. Math., 12 (1962), 481-498.
• T. Dube, Some ring-theoretic properties of almost $P$-frames, Algebra Universalis, 60(2) (2009), 145-162.
• T. Dube, Rings in which sums of $d$-ideals are $d$-ideals, J. Korean Math. Soc., 56(2) (2019), 539-558.
• T. Dube and L. Sithole, On the sublocale of an algebraic frame induced by the $d$-nucleus, Topology Appl., 263 (2019), 90-106.
• A. Facchini, C. A. Finocchiaro and G. Janelidze, Abstractly constructed prime spectra, Algebra Universalis, 83(1) (2022), 8 (38 pp).
• C. B. Huijsmans and B. de Pagter, On $z$-ideals and $d$-ideals in Riesz spaces I, Nederl. Akad. Wetensch. Indag. Math., 42(2) (1980), 183-195.
• C. B. Huijsmans and B. de Pagter, On $z$-ideals and $d$-ideals in Riesz spaces II, Nederl. Akad. Wetensch. Indag. Math., 42(4) (1980), 391-408.
• C. Jayaram, Baer ideals in commutative semiprime rings, Indian J. Pure Appl. Math., 15(8) (1984), 855-864.
• C. S. Manjarekar and A. N. Chavan, $z$-elements and $z_j$-elements in multiplicative lattices, Palest. J. Math., 8(1) (2019), 138-147.
• J. Martinez and E. Zenk, When an algebraic frame is regular, Algebra Universalis, 50(2) (2003), 231-257.
• G. Mason, Prime ideals and quotient rings of reduced rings, Math. Japon., 34(6) (1989), 941-956.
• C. J. Mulvey, &, Second topology conference (Taormina, 1984), Rend. Circ. Mat. Palermo (2) Suppl., 12 (1986), 99-104.
• S. B. Niefield and K. I. Rosenthal, Constructing locales from quantales, Math. Proc. Cambridge Philos. Soc., 104(2) (1988), 215-234.
• B. de Pagter, On $z$-ideals and $d$-ideals in Riesz spaces III, Nederl. Akad. Wetensch. Indag. Math., 43(4) (1981), 409-422.
• S. Safaeeyan and A. Taherifar, $d$-ideals, $fd$-ideals and prime ideals, Quaest. Math., 42(6) (2019), 717-732.
• H. Simmons, A framework for topology, Stud. Logic Found. Math., 96 (1978), 239-251.
• T. P. Speed, A note on commutative Baer rings, J. Austral. Math. Soc., 14 (1972), 257-263.
• M. Ward, Residuation in structures over which a multiplication is defined, Duke Math. J., 3(4) (1937), 627-636.
• M. Ward and R. P. Dilworth, Residuated lattices, Trans. Amer. Math. Soc., 45(3) (1939), 335-354.