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Probably a significant amount of matrix rings over division rings should just be covered under the Artin-Wedderburn theorem or semisimple ring or so.

Identification of rings as matrix rings is relatively interesting. For instance, if M_n(R) =~ M_m(S), must m=n or R=~S? A good reference for interesting items to add along these lines is Lam's Modules and Rings, the first chapter in the Morita equivalence section. I think this might be the main hope.

One could certainly add some trivial Morita corollaries to the properties list, but that only makes the article longer, not less stubby. One could link to the Jacobson density theorem which says rings with faithful primitive modules (like Matrix rings over division rings) are very nearly matrix rings over division rings.

One could link to Brauer group or Central simple algebra where matrix rings play the role of the identity element or trivial example, but again this is mostly an Artinian simple ring thing. JackSchmidt 23:34, 3 December 2007 (UTC)[reply]

The next posts are copied from Wikipedia talk:WikiProject Mathematics‎‎#Disambiguation help needed with Matrix algebra. D.Lazard (talk) 14:19, 15 February 2013 (UTC)[reply]

Matrix algebra has a large number of incoming links for which expert assistance is required. Please help if you can. Cheers! bd2412 T 03:00, 13 February 2013 (UTC)[reply]

IMO, matrix algebra should restored to a redirect to Matrix ring (its state of the beginning of 2005), and a hatnote {{Redirect}} should be added in the target.  Done D.Lazard (talk) 05:58, 13 February 2013 (UTC)[reply]

Substantiation? A confusion between the matrix algebra M_n(F ) and a matrix algebra (certain its subalgebra or, in other words, a faithful representation of some associative algebra in Fⁿ) does exist, isn’t it? Incnis Mrsi (talk) 10:57, 13 February 2013 (UTC)[reply]

Surely, such a confusion may exist. It must be resolved in matrix ring or in an article about matrix algebras, whichever name it should have. The dab page was not related with that confusion, but with the confusion between the subarea of mathematics and the algebraic structure(s). By the way, I have linked the hatnote in matrix ring to matrix (mathematics); another good choice would be linear algebra. I will add it to the hatnote. D.Lazard (talk) 11:23, 13 February 2013 (UTC)[reply]

I agree with the above comments, that neither Matrix algebra nor Adjoint representation should be disambiguation pages, per WP:TWODABS. I think Adjoint representation of a Lie group should be moved to Adjoint representation, and Matrix ring moved to Matrix algebra. I suppose we will need to start Requested Moves to accomplish such moves? Mark M (talk) 09:31, 13 February 2013 (UTC)[reply]

I would have thought that Matrix algebra should point to something like matrix multiplication -- ie how to do algebra with matrices, as the most entry-level and probably most commonly searched-for meaning of the term for the general reader. Of course a matrix algebra is a different thing, as is the notion of matrix algebras in general, but I suspect they may not be the most commonly sought (or even the most commonly linked) meanings of the term. Jheald (talk) 13:23, 13 February 2013 (UTC)[reply]

Hmm.. while I see your point, I'm not really convinced. The term "Matrix algebra" is definitely used in the sense of the article Matrix ring. Perhaps some people readers might use the term "Matrix algebra" to mean "algebra with matrices".. but matrix multiplication is linked from the first sentence of Matrix ring anyway, plus there is a hatnote further explaining the situation. So while you might be right, without convincing evidence that this is causing a problem, I don't see a reason to move. Also, the two concepts you are contrasting are in fact very closely related, of course. Mark M (talk) 14:14, 13 February 2013 (UTC)[reply]

I agree with Mark. I had a quick look on the links to matrix algebra. Most of them are well redirected to matrix ring. The other fall in two categories: in some mathematician biographies, "matrix algebra" could better be rewritten as "matrix algebra". In any case, for such non technical articles, such links seem rarely followed. The other category of links is related to software libraries; as, from a computing point of view, "matrix algebra" is not really different of "linear algebra", these links would be advantageously replaced by linear algebra. Thus, this a clear application of what BD2412 wrote above: "particularly if only one page contains an exact match to the title"; the only page containing an explicit definition of matrix algebra (in boldface) is matrix ring. D.Lazard (talk) 14:42, 13 February 2013 (UTC)[reply]

Looking at the discussion, I'm wondering about the "disambiguation" in the matrix ring article itself:

"Matrix algebra" redirects here. For the subarea of mathematics, see Matrix (mathematics) and Linear algebra.

If find that line rather odd and irritating. What exactly is the "subarea of mathematics"? And are we to believe that matrix ring itself is not mathematics (as the formulation may suggest to a (naive) reader)?

As far as the general problem is concerned I agree with Sławomir Biały, the style guide should not trump content. If for some reason no really appropriate article as a link target exists (yet), the the link target should remain the disambiguation page (for now). Enforcing a different not really appropriate link target just to comply the style guide is a no-go from my perspective.--Kmhkmh (talk) 15:17, 13 February 2013 (UTC)[reply]

I have rephrased the hatnote as follows:

This page is about a type of algebraic object. For algebraic operations on matrices see also Matrix (mathematics) and Linear algebra.

I had to do it by hand because we have the unusual situation that the alternative use is covered by the present article as well, just not at a level of abstraction appropriate for a typical reader looking for "matrix algebra" as a subfield of algebra. So I used "see also". Hans Adler 23:16, 14 February 2013 (UTC)[reply]

I do not agree: the new hatnote does not mean anything for readers that do not come to this article through the redirect "matrix algebra". Thus "redirect here" must be restored. I suggest: "Matrix algebra redirects here. For the algebraic theory of matrices see ..." I have not the time now, but unless somebody finds a better wording, I'll do this change later and transfer this discussion on the right talk page. D.Lazard (talk) 07:19, 15 February 2013 (UTC)[reply]

End of copied posts.

If n≥2, then M_n(R) is noncommutative and has lots of zero divisors, nilpotents, and idempotents. GeoffreyT2000 (talk) 15:00, 28 February 2015 (UTC)[reply]

In the Banach algebra example, some of the statements about infinite matrices look false to me. Does anyone know what they were intended to be? Is there a reference? Ebony Jackson (talk) 02:46, 26 December 2020 (UTC)[reply]