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Given a set X, what is the cardinality of the set of subsets of X which form a topology for X? In other words: what is the cardinality of the topologies of X? — Fly by Night ( talk) 01:07, 14 August 2010 (UTC)
Hi all - I'm working on the following problem and was looking for some guidance:
"Let R be a ring, and K a subring of R which is a field. Show that if R is an integral domain and then R is a field."
The problems I'm working on are from a course lectured at my university about 10 years ago, so I don't have an explicit definition of the notation, but I presume is the dimension of R as a vector space over K - however, my first query is, is it necessarily the case that if K and R are related as described above, then R can -definitely- be treated as a vector space over K? I don't need a proof or anything, I just want to check that's definitely the case.
Secondly and more pertinently, where should I start? I don't want a complete answer and indeed I'd rather not have one, I want to work through it myself - but where to begin? I considered writing an n-dimensional basis for R; and then expanding a general as , then finding a formula for the inverse, but I'm not sure if that's feasible or even a sensible way to go about the problem. Could anyone suggest a place to start?
Many thanks, 86.30.204.236 ( talk) 19:45, 14 August 2010 (UTC)
Ah, I have a habit of going in over the top, whoops! Thankyou very much, that's great 86.30.204.236 ( talk) 23:53, 14 August 2010 (UTC)
Mathematics desk | ||
---|---|---|
< August 13 | << Jul | August | Sep >> | August 15 > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
Given a set X, what is the cardinality of the set of subsets of X which form a topology for X? In other words: what is the cardinality of the topologies of X? — Fly by Night ( talk) 01:07, 14 August 2010 (UTC)
Hi all - I'm working on the following problem and was looking for some guidance:
"Let R be a ring, and K a subring of R which is a field. Show that if R is an integral domain and then R is a field."
The problems I'm working on are from a course lectured at my university about 10 years ago, so I don't have an explicit definition of the notation, but I presume is the dimension of R as a vector space over K - however, my first query is, is it necessarily the case that if K and R are related as described above, then R can -definitely- be treated as a vector space over K? I don't need a proof or anything, I just want to check that's definitely the case.
Secondly and more pertinently, where should I start? I don't want a complete answer and indeed I'd rather not have one, I want to work through it myself - but where to begin? I considered writing an n-dimensional basis for R; and then expanding a general as , then finding a formula for the inverse, but I'm not sure if that's feasible or even a sensible way to go about the problem. Could anyone suggest a place to start?
Many thanks, 86.30.204.236 ( talk) 19:45, 14 August 2010 (UTC)
Ah, I have a habit of going in over the top, whoops! Thankyou very much, that's great 86.30.204.236 ( talk) 23:53, 14 August 2010 (UTC)