On the intrinsic topology and some related ideals of

Authors:
O. A. S. Karamzadeh and M. Rostami

Journal:
Proc. Amer. Math. Soc. **93** (1985), 179-184

MSC:
Primary 54C40

DOI:
https://doi.org/10.1090/S0002-9939-1985-0766552-9

MathSciNet review:
766552

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The above topology is defined and studied on , the ring of real-valued continuous functions on a completely regular Hausdorff space . The minimal ideals and the socle of are characterized via their corresponding -filters. We observe that these ideals are -ideals and is discrete if and only if the socle of is a free ideal. It is also shown that for a class of topological spaces, containing all -spaces, the family of functions with compact support is identical with the intersection of the free maximal ideals of .

**[1]**F. F. Bonsall and J. Duncan,*Complete normed algebras*, Springer-Verlag, 1973. MR**0423029 (54:11013)****[2]**J. G. Brookshear,*On projective prime ideals in*, Proc. Amer. Math. Soc.**69**(1978), 203-204. MR**0470929 (57:10672)****[3]**W. W. Comfort,*On the Hewitt realcompactification of a product space*, Trans. Amer. Math. Soc.**131**(1968), 107-117. MR**0222846 (36:5896)****[4]**L. Gillman and M. Jerison,*Rings of continuous functions*, Springer-Verlag, 1976. MR**0407579 (53:11352)****[5]**L. Gillman,*Countably generated ideals in rings of continuous functions*, Proc. Amer. Math. Soc.**11**(1960), 660-666. MR**0156189 (27:6120)****[6]**O. Goldman,*A Wedderburn-Jacobson structure theorem*, J. Algebra**34**(1975), 64-73. MR**0366975 (51:3220)****[7]**R. C. Hayworth and R. A. McCoy,*Baire spaces*, Dissertationes Math.**141**(1977). MR**0431104 (55:4106)****[8]**M. Henriksen and M. Jerison,*The space of minimal prime ideals of a commutative ring*, Trans. Amer. Math. Soc.**115**(1965), 110-130. MR**0194880 (33:3086)****[9]**I. Kaplansky,*Topological rings*, Amer. J. Math.**69**(1947), 153-183. MR**0019596 (8:434b)****[10]**O. A. S. Karamzadeh,*Projective maximal right ideals of self-injective rings*, Proc. Amer. Math. Soc.**48**(1975), 286-288. MR**0360705 (50:13152)****[11]**O. A. S. Karamzadeh and M. Motamedi,*A note on rings in which every maximal ideal is generated by an idempotent*, Proc. Japan Acad. Ser. A Math. Sei.**58**(1982), 124-125. MR**664552 (83i:16015)****[12]**O. A. S. Karamzadeh,*On maximal right ideals which are direct summands*, Bull. Iran. Math. Soc.**18**(1983), 40-46. MR**738689 (85f:16037a)****[13]**O. A. S. Karamzadeh and M. Motamedi,*On the intersection of maximal right ideals which are direct summands*, Bull. Iran. Math. Soc.**18**(1983), 47-54. MR**738690 (85f:16037b)****[14]**O. A. S. Karamzadeh,*On the classical Krull dimension of rings*, Fund. Math.**117**(1983). MR**719833 (84k:16030)****[15]**C. W. Kohls,*Ideals in rings of continuous functions*, Fund. Math.**45**(1975), 28-50. MR**0102731 (21:1517)****[16]**J. Lambek,*Rings and modules*, Blaisdell, New York, 1966.**[17]**M. Mandelker,*Supports of continuous functions*, Trans. Amer. Math. Soc.**156**(1971), 73-83. MR**0275367 (43:1124)****[18]**G. De Marco,*On the countably generated**-ideals of*, Proc. Amer. Math. Soc.**31**(1972), 574-576. MR**0288563 (44:5760)****[19]**M. Rajagopalan and R. F. Wheeler,*Sequential compactness of**implies a compactness property for*, Canad J. Math.**28**(1976), 207-210. MR**0405341 (53:9135)****[20 S]**M. Robinson,*The intersection of the free maximal ideals in a complete space*, Proc. Amer. Math. Soc.**17**(1966), 468-469. MR**0188974 (32:6401)****[21]**D. Rudd,*-ideals and**-ideals in rings of continuous functions*, Fund. Math.**88**(1970), 53-59. MR**0380716 (52:1613)****[22]**W. Rudin,*Continuous functions on compact spaces without perfect subsets*, Proc. Amer. Math. Soc.**8**(1957), 39-42. MR**0085475 (19:46b)****[23]**R. F. Wheeler,*The strict topology for**-spaces*, Proc. Amer. Math. Soc.**41**(1973), 466-467. MR**0341048 (49:5798)****[24]**O. Zariski and P. Samuel,*Commutative algebra*, vol. 1, Van Nostrand, New York, 1958. MR**0090581 (19:833e)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
54C40

Retrieve articles in all journals with MSC: 54C40

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1985-0766552-9

Keywords:
Isolated maximal ideals,
intrinsic topology,
-space,
minimal ideal,
socle,
real pseudo-finite space,
-ideal,
-ideal

Article copyright:
© Copyright 1985
American Mathematical Society