For an associative ring \(R\) with identity let \(J(R) \) be the Jacobson radical of \(R\) and \(N(R)\) the non-left-invertible elements of \(R\). \(R\) has stable range one if for any \(a,b\in R\) such that \(Ra+Rb=R\) there exists \(y\in R\) such that \(a+yb\) is a unit. \(R\) is (strongly) \(\pi\)-regular if for every \(a\in R\) there exists a positive integer \(n\), depending on \(a\), such that (\(a^n\in a^{n+1}R\)) \(a^n\in a^nRa^n\) and \(R\) is semilocal if \(R/J(R)\) is a left Artinian ring. \(R\) is \(J\)-semisimple if its Jacobson radical is zero. \(R\) is decomposable if it contains a central idempotent distinct from \(0\) and \(1\) and indecomposable otherwise. The collection of all \(n\times n\) matrices over a ring \(R\) is denoted by \(M_n(R)\) and the rings of row finite and column finite matrices over \(R\) are denoted by \(CFM_n(R)\) and \(RFM_n(R)\), respectively, where \(n\) can be any finite or infinite cardinal number. A ring \(R\) is called \(J^*\)-ring if for every \(x\in R\), either \(x\in J(R)\) or \(x=a+b\), where \(a\) is left-invertible and \(b\) is non-left-invertible. In this paper the authors study the generalization \(J^*(R)=\{x\in N(R):x+N(R)\subseteq N(R)\}\) of the Jacobson radical as well as the rings \(R\) such that \(J^*(R)=J(R)\). In particular, they show that \(J^*(R)\) is an associative ring, \(J^*(\overline R)=\overline{J^*(R)}\), where \(\overline R=R/J(R)\) and for any direct product \(\prod_{i\in I}R_i\) of rings we have \(J^*(\prod_{i\in I}R_i)=\prod_{i\in I}J^*(R_i)\). They also show that for any finite or countably infinite cardinal \(n\) we have (i) \(J^*(CFM_n(R))\subseteq CFM_n(J^*(R))\) and (ii) \(J^*(RFM_n(R))\subseteq RFM_n(J^*(R))\) and that these inclusions may be strict. Moreover, the authors prove that \(I\subseteq J^*(R)\) if and only if \(I\subseteq J(R)\), for any ideal \(I\) of \(R\). Also the authors show that the following conditions are equivalent: (i) \(R\) is a \(J^*\)-ring, (ii) \(J^*(R)=J(R)\), (iii) \(J^*(R)\) is a left ideal, (iv) \(\overline R=R/J(R)\) is a \(J^*\)-ring, (v) the split-null (or trivial) extension \(S(R,M)\) of a unital \((R,R)\)-bimodule \(M\) by \(R\) is a \(J^*\)-ring, (vi) every upper (lower) triangular matrix ring \(UTM_n(R)\) (\(LTM_n(R)\)) is a \(J^*\)-ring for every finite or countably infinite cardinal number \(n\). Finally the authors show that \(J^*(R)=J(R)\) for the following classes of rings: semilocal rings, stable range one rings, rings which are generated by their units, left Artinian rings, \(CFM_n(R)\), where \(R\) is a \(J^*\)-ring, \(J(CFM_n(R))=CFM_n(J(R))\) and \(n\) is any finite or countably infinite cardinal number. Similarly, under the same conditions, \(RFM_n(R)\) is also a \(J^*\)-ring and also every ring whose \(J\)-semisimple factor-rings are \(J^*\)-rings.