Maximal regular right ideal space of a primitive ring

Authors:
Kwangil Koh and Jiang Luh

Journal:
Trans. Amer. Math. Soc. **170** (1972), 269-277

MSC:
Primary 16A20

DOI:
https://doi.org/10.1090/S0002-9947-1972-0304413-5

MathSciNet review:
0304413

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Abstract: If is a ring, let be the set of maximal regular right ideals of and be the lattice of right ideals. For each , define . We give a topology to by taking as a subbase. Let be a right primitive ring. Then is the union of two proper closed sets if and only if is isomorphic to a dense ring with nonzero socle of linear transformations of a vector space of dimension two or more over a finite field. is a Hausdorff space if and only if either is a division ring or modulo its socle is a radical ring and is isomorphic to a dense ring of linear transformations of a vector space of dimension two or more over a finite field.

**[1]**A. Białynicki-Birula, J. Browkin, and A. Schinzel,*On the representation of fields as finite unions of subfields*, Colloq. Math.**7**(1959), 31–32. MR**0111739**, https://doi.org/10.4064/cm-7-1-31-32**[2]**Nathan Jacobson,*Structure of rings*, American Mathematical Society Colloquium Publications, Vol. 37. Revised edition, American Mathematical Society, Providence, R.I., 1964. MR**0222106****[3]**R. S. Pierce,*Modules over commutative regular rings*, Memoirs of the American Mathematical Society, No. 70, American Mathematical Society, Providence, R.I., 1967. MR**0217056****[4]**R. G. Swan,*Algebraic 𝐾-theory*, Lecture Notes in Mathematics, No. 76, Springer-Verlag, Berlin-New York, 1968. MR**0245634**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1972-0304413-5

Keywords:
Maximal regular right ideals,
socle,
reducible spaces,
Hausdorff spaces,
support

Article copyright:
© Copyright 1972
American Mathematical Society