Bulletin of the Australian Mathematical Society Bull. Austral. Math. Soc. 0004-9727 ALNBAB 2005 72 3 10.wxyz/CV72P3 http://www.austms.org.au/Publ/Bulletin/V72P3/ Halina France-Jackson 13 February 2006 2006 2 13 403 406 723-5200-FranceJackson-2005 10.wxyz/C2005V72P3p403 http://www.austms.org.au/Publ/Bulletin/V72P3/723-5200-FranceJackson/ A radical $\alpha $ in the universal class of all associative rings is called matric-extensible if for all natural numbers $n$ and all rings $A$, $A\in \alpha $ if and only if $M_{n}( A) \in \alpha $, where \ $M_{n}( A) $ denotes the $n\times n$ matrix ring with entries from $A$. We show that there are no coatoms, that is, maximal elements in the lattice of all matric-extensible radicals of associative rings. 16N80 V.A. Andrunakievich and Yu. M. Ryabukhin (in Russian) Nauka V.A. Andrunakievich and Yu. M. Ryabukhin; \textit{Radicals of algebras and structure theory}, (in Russian) (Nauka, Moscow, 1979). G.L. Booth and H. France-Jackson Acta Math. Hungar. MR2011471 G.L. Booth and H. France-Jackson; On the lattice of matric-extensible radicals, \textit{Acta Math. Hungar.} \textbf{101} (2003), pp.~163--172. B.J. Gardner and P.N. Stewart Proc. Edinburgh Math. Soc. MR1115644 B.J. Gardner and P.N. Stewart; Prime essential rings, \textit{Proc. Edinburgh Math. Soc.} \textbf{34} (1991), pp.~241--250. B.J. Gardner and R. Wiegandt Marcel Dekker MR2015465 B.J. Gardner and R. Wiegandt; \textit{Radical theory of rings} (Marcel Dekker, New York, 2004). E.R. Puczylowski Glasgow Math. J. MR641621 E.R. Puczylowski; Hereditariness of strong and stable radicals, \textit{Glasgow Math. J.} \textbf{23} (1982), pp.~85--90. E.R. Puczylowski and E. Roszkowska Comm. Algebra MR1154396 E.R. Puczylowski and E. Roszkowska; On atoms and coatoms of lattices of radicals of associative rings, \textit{Comm. Algebra} \textbf{20} (1992), pp.~955--977. R.L. Snider Pacific. J. Math. MR308175 R.L. Snider; Lattices of radicals, \textit{Pacific. J. Math.} \textbf{40} (1972), pp.~207--220.