For any finite directed graph E and any field K we show that the Cohn path algebra C K (E) has the Invariant Basis Number property, moreover Invariant Matrix Number property. 2010 Mathematics Subject Classification. Primary 16S99 Secondary 05C25.

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The necessary and sufficient conditions are given to assure the existence of a maximal ideal in L and also the necessary and sufficient conditions on the graph that assure that every ideal is contained in a maximal ideal are given. It is shown that if a maximal ideal M of L is nongraded, then the largest graded ideal in M , namely gr(M) , is also maximal among the graded ideals of L. Moreover, if L has a unique maximal ideal M , then M must be a graded ideal. The necessary and sufficient conditions on the graph for which every maximal ideal is graded are discussed.

Prüfer domains and subclasses of integral domains such as Dedekind domains admit characterizations by means of the properties of their ideal lattices. Interestingly, a Leavitt path algebra L, in spite of being noncommutative and possessing plenty of zero divisors, seems to have its ideal lattices possess the characterizing properties of these special domains.In [8] it was shown that the ideals of L satisfy the distributive law, a property of Prüfer domains and that L is a multiplication ring, a property of Dedekind domains. In this paper, we first show that L satisfies two more characterizing properties of Prüfer domains which are the ideal versions of two theorems in Elementary Number Theory, namely, for positive integers a, b, c, gcd(a, b) · lcm(a, b) = a · b and a · gcd(b, c) = gcd(ab, ac). We also show that L satisfies a characterizing property of almost Dedekind domains in terms of the ideals whose radicals are prime ideals. Finally, we give necessary and sufficient conditions under which L satisfies another important characterizing property of almost Dedekind domains, namely the cancellative property of its non-zero ideals. 0 2010 Mathematics Subject Classification: 16D25; 13F05

We give the necessary and sufficient condition for a separated Cohn–Leavitt path algebra of a finite digraph to have Invariant Basis Number (IBN). As a consequence, separated Cohn path algebras have IBN. We determine the non-stable K-theory of a corner ring in terms of the non-stable K-theory of the ambient ring. We give a necessary condition for a corner algebra of a separated Cohn–Leavitt path algebra of a finite graph to have IBN. We provide Morita equivalent rings which are non-IBN, but are of different types.