Dedekind-finiteRings

A ring $R$ is said to be Dedekind-finite if for $a,b\in R$, $ab=1$ implies $ba=1$. Certainly if $R$ is commutative, then $R$ is Dedekind-finite. Other examples include

The endormophism ring $\mathrm{End}_k(V)$ is Dedekind-finite where $V$ is a finite dimensional vector space over the field $k$. [Linear algebra]
An algebraic algebra over a field $k$ is Dedekind-finite. [Ex. 1.12 (p. 23), L]
A domain is Dedekind-finite. [Ex. 1.4 (p. 22), L]
A semisimple ring is Dedekind-finite. [Ex. 3.10 (p. 46), L]
A semilocal ring is Dedekind-finite. [Proposition 20.8 (p. 298), L] (Note. Any local ring is semilocal.)
Let $R$ be a ring such that $R/\mathrm{rad}(R)$ contains no infinite direct sum of nonzero right ideals. Then $R$ is Dedekind finite. [Corollary 21.27 (p. 318), L]
Let $R$ be a ring with left stable range 1. Then $M_n(R)$ is Dedekind-finite for all $n\geq 1$. [Theorem 20.13 (p. 301), L]

Example. A non-Dedekind-finite ring [p. 4, L]. Let $V$ be a vector space over a field $k$ with countable infinite basis $\{e_i\mid i=1,2,\dots\}$. Let $R=\mathrm{End}_k(V)$. Let $a,b\in R$ be given by $a(e_1)=0$ and $a(e_i)=e_{i-1}$ for all $i\geq 2$, and $b(e_i)=e_{i+1}$ for all $i$. Then $ab=1$ but $ba\not=1$.

Definition. A ring $R$ is called weakly $n$-finite if $M_n(R)$ is Dedekind-finite. Thus, a ring is weakly $1$-finite if and only if it is Dedekind-finite.

Note 1. Weakly $1$-finite ring may not be weakly $2$-finite. As indicated in [Section 3, C], an example is given in [S]. Cohn also give an example in [C].

Note 2. In the article [HL], many properties of Dedekind-finite regular rings are discussed. The word "finite" in the titile of the article refers to "Dedekind-finite"!

Example. Using $a$ and $b$ in the above example of non-Dedekind-finite ring, W Holubowski constructed an upper triangular matrix whose inverse matrix is lower triangular. So take $c\in R$ such that $c(e_1)=e_1$ and $c(e_i)=0$ for all $i\geq 2$. Let $A=\begin{pmatrix}a&0\\c&b\end{pmatrix}$ and $B=\begin{pmatrix}b&c\\0&a\end{pmatrix}$. Then $AB=BA=I$.

Note 3. If $R$ is a Dedekind-finite ring and $S=M_n(R)$, $n\geq 2$, or $n$ is countably infinite, then the inverse of a lower (upper) triangular matrix in $S$ is lower (upper) triangular. Consequently, the ring $L_n(R)$ ($U_n(R)$) of lower (upper) triangular matrices is Dedekind-finite.

Note 4. In the abstract of a talk by Patricio [P], it is stated that if $R$ is a ring with involution, then all lower triangular invertible matrices have lower triangular inverses if and only if the ring R is Dedekind-finite.

Question. Is the statement in Note 4 true? If so, is involution necessary?

Problem. Study similar problems in nearrings.

References

[C] P. M. Cohn, On $n$-simple rings. Algebra univers. 53 (2005) 301–-305.
[H] W. Holubowski, An inverse matrix of an upper triangular matrix can be lower triangular. Discussiones Mathematicae. General Algebra and Applications 22 (2002), 161--166.
[HL] R. E. Hartwig and J. Luh, On finite regular rings. Pacific J. Math. 69 (1977), 73--95.
[L] T. Y. Lam. A first course in noncommutative rings, 2nd Edition. Graduate Texts in Mathematics 131. Springer-Verlag, New York 2001.
[P] P. Patricio, Lower triangular matrices with lower triangular Moore-Penrose inverses. All Ireland Algebra Days 2001, May 16-19, 2001, Queen's University Belfast Belfast, Northern Ireland, UK. (Abstract)
[S] J. C. Shepherdson, Inverses and zero divisors in matrix rings, Proc. London Math. 1 (1951), 71–-85.