The structure of the plactic algebra K[M] of rank 2 or 3 is studied. It is shown that these algebras are semiprimitive. Moreover, the prime spectrum is completely described in the rank 2 case. Also it is proved that the plactic algebra of rank n ≥ 4 is not semiprime and the plactic algebra of rank 3 is not prime. © 2004 Elsevier Inc. All rights reserved.