Gabriel Quotient Rings
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866913785029066752 |
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| author | Wangneo, C L |
| author_facet | Wangneo, C L |
| contents | In this paper we prove the following theorem. Let R be a prime Noetherian ring with krull dimension |R| = n where n is a positive integer. Let Q be the Goldie quotient ring of R. For a fixed positive integer m < n, let xm be the set of all prime ideals of R such that krull dimension R/p equals m. Call xm the set of m-full prime ideals of R. Let Cm be the set of elements c of R With krull dimension R/cR less than m. Call g as the m-gabriel filter, if g is the family of right ideals I of R with krull dimension R/I less than m. We construct an extension ring R(m) of R having the following properties (i) R(m) is a subring of Q with identity element 1 of R. (ii) If u(R(m)) is the set of units of R(m) then u(R(m) intersection R equals the set cm. (iii) For a full set of m-prime ideals of R the set cm is a right ore set of R . We call R(m) as the m-Gabriel quotient ring of R. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_13186 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Gabriel Quotient Rings Wangneo, C L Rings and Algebras In this paper we prove the following theorem. Let R be a prime Noetherian ring with krull dimension |R| = n where n is a positive integer. Let Q be the Goldie quotient ring of R. For a fixed positive integer m < n, let xm be the set of all prime ideals of R such that krull dimension R/p equals m. Call xm the set of m-full prime ideals of R. Let Cm be the set of elements c of R With krull dimension R/cR less than m. Call g as the m-gabriel filter, if g is the family of right ideals I of R with krull dimension R/I less than m. We construct an extension ring R(m) of R having the following properties (i) R(m) is a subring of Q with identity element 1 of R. (ii) If u(R(m)) is the set of units of R(m) then u(R(m) intersection R equals the set cm. (iii) For a full set of m-prime ideals of R the set cm is a right ore set of R . We call R(m) as the m-Gabriel quotient ring of R. |
| title | Gabriel Quotient Rings |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2308.13186 |