Geodesic
Implications of a system of linear equations over a module
Diskretnaya Matematika, Tome 19 (2007) no. 1, pp. 133-140 
V. P. Elizarov. Implications of a system of linear equations over a module. Diskretnaya Matematika, Tome 19 (2007) no. 1, pp. 133-140. http://geodesic.mathdoc.fr/item/DM_2007_19_1_a13/
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     title = {Implications of a~system of linear equations over a~module},
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     url = {http://geodesic.mathdoc.fr/item/DM_2007_19_1_a13/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

We describe the class $L(R)$ of all left modules over a ring $R$ such that for any matrix $D$ over $R$ and any solvable system of equations $$ F\eta^\downarrow=\gamma^\downarrow $$ over a module from $L(R)$ the system of equations $$ A\xi^\downarrow=\beta^\downarrow $$ is its $D$-implication if and only if $$ T(F,\gamma^\downarrow)=(AD,\beta^\downarrow) $$ for some matrix $T$. If $R$ is a quasi-Frobenius ring, then $L(R)$ contains the subclass of all faithful $R$-modules. A criterion for a system of equations over a module from $L(R)$ to be definite is obtained.

[1] Glukhov M. M., Elizarov V. P., Nechaev A. A., Algebra, t. 1, 2, Gelios ARV, Moskva, 2003

[2] Elizarov V. P., “Sistemy lineinykh uravnenii nad kvazifrobeniusovymi koltsami”, Fundament. i prikl. matem., 1:2 (1995), 535–539 | MR | Zbl

[3] Elizarov V. P., “Opredelennye sistemy lineinykh uravnenii nad koltsami”, Fundament. i prikl. matem., 4:4 (1998), 1307–1313 | MR | Zbl

[4] Elizarov V. P., “Metody uproscheniya sistem lineinykh uravnenii nad modulyami. I”, Trudy po diskretnoi matematike, 7 (2003), 56–74 | MR

[5] Kash F., Moduli i koltsa, Mir, Moskva, 1981

[6] Nechaev A. A., “Konechnye kvazifrobeniusovy moduli, prilozheniya k kodam i lineinym rekurrentam”, Fundament. i prikl. matem., 1:1 (1995), 229–254 | MR | Zbl

[7] Hall M., “A type of algebraic closure”, Ann. Math., 40 (1939), 360–369 | DOI | MR | Zbl