You might be interested in this paper by Keith Conrad, arguing that the "correct" definition of a ring should require a multiplicative identity:

http://www.math.uconn.edu/~kconrad/blur ... ngdefs.pdf

## Search found 10 matches

- Wed May 04, 2011 5:56 am UTC
- Forum: Mathematics
- Topic: Must a ring have multiplicative identity?
- Replies:
**8** - Views:
**2792**

- Wed Feb 02, 2011 11:56 am UTC
- Forum: Mathematics
- Topic: Math Books
- Replies:
**379** - Views:
**266297**

### Re: Math Books

So, after doing Measure Theory and such things, I want more analysis stuff. I have already gone through Royden, now I don't know where to go next. What kind of book would be good if I want to learn about things like Sobolev spaces? And other things in analysis? I know functional analysis is a place...

- Sat Dec 18, 2010 8:51 am UTC
- Forum: Mathematics
- Topic: ln(x) or log(x)?
- Replies:
**63** - Views:
**8948**

### Re: ln(x) or log(x)?

Once you get past introductory calculus classes, there is pretty much universal agreement in math that log means the "natural" or "base-e" logarithm. In fact, I don't recall ever seeing "ln" in any math textbook beyond the calculus level.

- Tue Dec 14, 2010 8:04 am UTC
- Forum: Mathematics
- Topic: Primes vs. Natural Numbers
- Replies:
**9** - Views:
**1705**

### Re: Primes vs. Natural Numbers

For yet another way to think about "how many" primes there are among the natural numbers, recall that [imath]\sum 1/p[/imath] diverges. So in this sense there are many more primes than other density-zero subsets.

- Sun Dec 12, 2010 5:44 pm UTC
- Forum: Mathematics
- Topic: I have a measure theory final on tuesday
- Replies:
**24** - Views:
**2341**

### Re: I have a measure theory final on tuesday

This post on Terry Tao's blog gives problem-solving strategies for real variables problems.

http://terrytao.wordpress.com/2010/10/2 ... trategies/

http://terrytao.wordpress.com/2010/10/2 ... trategies/

- Sat Dec 11, 2010 9:39 pm UTC
- Forum: Mathematics
- Topic: Cantor Diagonalization Formula
- Replies:
**11** - Views:
**2705**

### Re: Cantor Diagonalization Formula

You'll want to check me on this, but I believe the map f: N\times N \rightarrow N given by f(m,n) = \frac{1}{2} (m^2 + n^2 + m(2n-3) - n + 2) is a bijection. On a side note, have you looked at the Calkin-Wilf Tree? That gives a very interesting bijection with connections to n...

- Tue Nov 23, 2010 8:11 am UTC
- Forum: Mathematics
- Topic: Fractal-like circle thingy
- Replies:
**26** - Views:
**2673**

### Re: Fractal-like circle thingy

If you find the arc length by integrating over the path with respect to the 1-dimensional Hausdorff measure in R

^{2}, this is a perfect example of why Fatou's Lemma gives an inequality and not an equality.- Tue Nov 23, 2010 4:14 am UTC
- Forum: Mathematics
- Topic: Quick Q about LaPlace transforms
- Replies:
**4** - Views:
**576**

### Re: Quick Q about LaPlace transforms

It certainly looks like a typo. A convolution wouldn't make sense here, as e^{2x} \cos(3(t-x)) \notin L^1(R,x) . And besides, if you take the inverse Laplace transform of the given answer, you just get -\frac{1}{10} e^{-3t} - \frac{2}{5} e^{2t} + \frac{1}{2} e^{3t} which...

- Mon Nov 22, 2010 11:27 pm UTC
- Forum: Mathematics
- Topic: Entire Function That Agrees with Log on Natural Numbers?
- Replies:
**25** - Views:
**3406**

### Re: Entire Function That Agrees with Log on Natural Numbers?

As an aside, I wonder why the conditions on the two theorems are stated as |z_n| \to \infty for one and discrete for the other... these seem equivalent to me, as any discrete set can only have finitely many values in any compact subset, in particular for the set |z| \le M , and any sequence whose m...

- Mon Nov 22, 2010 10:57 am UTC
- Forum: Mathematics
- Topic: Entire Function That Agrees with Log on Natural Numbers?
- Replies:
**25** - Views:
**3406**

### Re: Entire Function That Agrees with Log on Natural Numbers?

Given a set \{z_n\} of complex numbers with \lvert z_n \rvert \rightarrow \infty , it is always possible to find an entire function with a prescribed value at each z_n . This is a consequence of the Weierstrass Factorization Theorem and the Mittag-Leffler Theorem (see Ahlfors or any other complex va...