Distributive and multiplication modules. (English) Zbl 1228.13015
In a paper by A. Barnard [J. Algebra 71, 174–178 (1981; Zbl 0468.13011)], \(M\) is called a multiplication module over a commutative ring \(R\) with identity means here that every submodule has the form \(IM\) for some ideal \(I\) of \(R\). An \(R\)-module \(M\) is said to be a distributive module if for all submodules \(N\), \(K\), and \(L\) of \(M\) we have, \(N+(K\cap L)=(N+K)\cap (N+L)\).
In the paper under review the authors show that every distributive module satisfies the radical formula. Also they provide a proof of the prime avoidance theorem for distributive modules. In addition, they study the question when a multiplication module becomes flat.
In the paper under review the authors show that every distributive module satisfies the radical formula. Also they provide a proof of the prime avoidance theorem for distributive modules. In addition, they study the question when a multiplication module becomes flat.
Reviewer: Siamak Yassemi (Tehran)
MSC:
| 13C10 | Projective and free modules and ideals in commutative rings |
| 13C11 | Injective and flat modules and ideals in commutative rings |
| 13C13 | Other special types of modules and ideals in commutative rings |
Citations:
Zbl 0468.13011References:
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