most recent 30 from http://mathoverflow.net 2013-05-24T22:44:52Z http://mathoverflow.net/feeds/question/22676 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22676/terminology-about-ring-algebra-in-abstract-algebra-and-measure-theory zzzhhh 2010-04-27T04:08:06Z 2010-08-03T03:17:52Z

<p>Both in abstract algebra and measure theory is there term ring/algebra, but their definition are different and we can not deduce one from the other, the only requirement in definition they share is that they be closed under two operations: addition and multiplication in abstract algebra while difference and union in measure theory. Why we use the same term for these two different notions in two different branches of mathematics? Is there any connection between them? Thanks! </p> <p>To be more specific, my intention is to compare "ring in measure theory" with "ring in abstract algebra", not "ring" with "algebra". I use notation "ring/algebra" in the post because the term algebra has the same situation, that is, compare "algebra in measure theory" with "algebra in abstract algebra".</p>

http://mathoverflow.net/questions/22676/terminology-about-ring-algebra-in-abstract-algebra-and-measure-theory/22679#22679 Gjergji Zaimi 2010-04-27T04:38:30Z 2010-04-27T18:25:27Z

<p>A <a href="http://en.wikipedia.org/wiki/Ring_of_sets" rel="nofollow">ring of sets</a> is a ring(usual definition) with the operations intersection(multiplication) and symmetric difference(addition). A <a href="http://en.wikipedia.org/wiki/Sigma_ring" rel="nofollow">sigma ring</a> is a special kind of a ring of sets. Now a <a href="http://en.wikipedia.org/wiki/%CE%A3-algebra" rel="nofollow">sigma algebra</a> <strike>(which would possibly more appropriately be called sigma field) is a sigma ring where every element has a complement (multiplicative inverse)</strike> is a sigma ring with unity (equivalent to every element has a complement),thus it is a Boolean algebra with respect to intersection and union, and a Boolean ring with respect to intersection and symmetric difference. Every Boolean ring is an algebra over $\mathbb F_2$(thank you Mark Meckes for the correction). I hope this connection is enough to justify the use of these terms.</p>

http://mathoverflow.net/questions/22676/terminology-about-ring-algebra-in-abstract-algebra-and-measure-theory/22680#22680 Pete L. Clark 2010-04-27T04:39:14Z 2010-04-27T04:39:14Z

<p>There is a mathematical relationship between the two concepts: given an algebra of sets, one can define a commutative ring by taking addition to be symmetric difference and multiplication to be intersection. This ring is a <strong>Boolean ring</strong>, i.e., $x^2 = x$ for all elements $x$. Conversely, the Stone Representation Theorem says that every Boolean ring can be viewed as the ring associated to some algebra of sets. See e.g. Section 4.5</p> <p><a href="http://math.uga.edu/~pete/integral.pdf" rel="nofollow">http://math.uga.edu/~pete/integral.pdf</a></p> <p>for more detailed statements and proofs. </p>

http://mathoverflow.net/questions/22676/terminology-about-ring-algebra-in-abstract-algebra-and-measure-theory/34322#34322 jackie boy 2010-08-03T03:17:52Z 2010-08-03T03:17:52Z

<p>For what it's worth, a ring is an algebra over the integers.</p>