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Let R be a commutative ring with identity 1 = 0. We say that a ring is a chained ring if either x j y or y j x for all x; y 2 R. The set of the zero-divisors of a ring <br /> <br /> <br /> is Z(R), and Nil(R) its ideal of nilpotent elements. The zero-divisor graph of R is T(R) =...
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| 主要作者: | |
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| 格式: | Final Project |
| 語言: | Indonesia |
| 在線閱讀: | https://digilib.itb.ac.id/gdl/view/15527 |
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| 機構: | Institut Teknologi Bandung |
| 語言: | Indonesia |
| 總結: | Let R be a commutative ring with identity 1 = 0. We say that a ring is a chained ring if either x j y or y j x for all x; y 2 R. The set of the zero-divisors of a ring <br />
<br />
<br />
is Z(R), and Nil(R) its ideal of nilpotent elements. The zero-divisor graph of R is T(R) = Z(R) n (0), with distinct vertices x and y adjacent if and only if xy = 0. <br />
<br />
<br />
In this manuscript, we show that the diameter of the zero-divisor graph of a chained ring is diam (T(R)) < 2. The diameter of the zero-divisor graph of a ring such that <br />
<br />
<br />
the prime ideals of R contained in Z(R) are linearly ordered is 2. |
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