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  "body": "\n## Definition\n\nLet $K$ be a field and let $A$ be a finite-dimensional [[associative algebra|3CTW1P4HBP]] over $K$.\nLet $P_1, \\dots, P_n$ be representatives of the isomorphism classes of [[indecomposable|E2P29GCD9T]] [[projective|BFFA03GE3V]] left $A$-modules.\nLet $S_i = P_i / JP_i$, where $J$ is the [[Jacobson radical|XA0DCNH7WY]] of $A$; then $S_1, \\dots, S_n$ represent the isomorphism classes of [[simple|G00C22AEDH]] left $A$-modules.\nThe *Cartan matrix* of $A$ is the $n \\times n$ integer matrix $C = (c_{ij})$.\nHere $c_{ij}$ is the number of times $S_j$ occurs among the [[composition factors|0PHTC93M58]] of $P_i$.\n\n## Remarks\n\nExample: $A = K[x]/(x^2)$ has the single indecomposable projective $P_1 = A$, with composition series $0 \\subset (x) \\subset A$.\nBoth factors are the simple module $K$, so $C = (2)$.\nExample: for the algebra $T_2(K)$ of upper triangular $2 \\times 2$ matrices, $P_1 = K E_{11}$ is simple and $P_2 = K E_{12} + K E_{22}$ has the factors $S_2$ on top and $S_1$ below, so $C = \\begin{pmatrix} 1 \u0026 0 \\\\ 1 \u0026 1 \\end{pmatrix}$.\nNon-example: the polynomial ring $K[x]$ is not finite-dimensional, and its projective module $K[x]$ has no composition series, so it has no Cartan matrix in this sense.\nSome authors use the transpose of $C$.\nIf $K$ is a splitting field for $A$, for example algebraically closed, then $c_{ij} = \\dim_K \\Hom_A(P_j, P_i)$.\nFor the group algebra $KG$ of a finite group over a splitting field of characteristic $p$, Brauer showed that $C = D^{T} D$, where $D$ is the decomposition matrix, so $C$ is symmetric.\n",
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