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Roughness in rings. (English) Zbl 1072.16042

Let \(I\) be an ideal of a ring \(R\). Let \(P(R)\) denote the power set of \(R\). Define \(\underline{Apr}_I\colon P(R)\to P(R)\) and \(\overline{Apr}_I\colon P(R)\to P(R)\) as follows: \(\forall X\in P(R)\), \(\underline{Apr}_I(X)=\{x\in R\mid x+I\subseteq X\}\) and \(\overline{Apr}_I(X)=\{x\in R\mid(x+I)\cap X\neq\emptyset\}\). Then \(\underline{Apr}_I(X)\) and \(\overline{Apr}_I(X)\) are called the lower and upper approximations of \(X\), respectively, with respect to the ideal \(I\). This definition is consistent with the usual definitions in rough set theory since \(I\) defines an equivalence relation \(\sim\) on \(R\), i.e., \(a\sim b\Leftrightarrow a-b\in I\) and \(x+I\) is an equivalence class of \(\sim\).
The author then proves basic properties of lower and upper approximations on \(R\). For example, let \(A\) and \(B\) be nonempty subsets of \(R\). Then the author shows that (1) \(\overline{Apr}_I(A)\bullet\overline{Apr}_I(B)=\overline{Apr}_I(A\bullet B)\); (2) \(\overline{Apr}_I(A)+\overline{Apr}_I(B)=\overline{Apr}_I(A+B)\); (3) \(\underline{Apr}_I(A)\bullet\underline{Apr}_I(B)\subseteq\underline{Apr}_I(A \bullet B)\); (4) \(\underline{Apr}_I(A)+\underline{Apr}_I(B)\subseteq \underline{Apr}_I(A+B)\).
Let \(J\) be an ideal (subring) of \(R\). The author shows that \(\overline{Apr}_I(J)\) and \(\underline{Apr}_I(J)\) are ideals (subrings) of \(R\). Let \(I\) and \(J\) be ideals of \(R\) and \(K\) a subring of \(R\). The author shows that \(\overline{Apr}_I(K)\bullet\overline{Apr}_J(K)\subseteq \overline{Apr}_{(I+J)}(K)\); \(\underline{Apr}_I(K)\bullet\underline{Apr}_J(K)= \underline{Apr}_{(I+J)}(K)\); \(\overline{Apr}_I(K)+\overline{Apr}_J(K)= \overline{Apr}_{(I+J)}(K)\); \(\underline{Apr}_I(K)+\underline{Apr}_J(K)= \underline{Apr}_{(I+J)}(K)\).
The author examines the homomorphic images of lower and upper approximations of sets. He then studies fuzzy sets, fuzzy rough sets and fuzzy rough ideals with respect to lower and upper approximations.

MSC:

16Y99 Generalizations
16D25 Ideals in associative algebras
03E72 Theory of fuzzy sets, etc.
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