For finite-dimensional algebras over an algebraically closed field, the second Brauer-Thrall conjecture was proved by several authors (first by Nazarova and Rojter, then by Bretscher and Todorov, Bautista, and Fischbacher, etc.). But for Artin algebras over a commutative ring, this conjecture is still open. One of the main results of the paper is to prove the second Brauer-Thrall conjecture for an Artin algebra \(A\) such that \(\text{rad}^ \infty(M,M)=0\) for all indecomposable modules \(M\) in \(\text{mod }A\), the category of all finitely generated \(A\)-modules. Such an algebra is called a loop-finite algebra. The author shows also that the class of minimal representation infinite loop-finite Artin algebras is just the class of tame concealed Artin algebras. Several equivalent conditions of representation-infinite algebras are also presented. As a consequence of these results the author proves that if an Artin algebra \(A\) is such that the endomorphism ring of any indecomposable module in \(\text{mod }A\) is a division ring, then \(A\) is representation-finite.