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A Construction in General Radical Theory

  • Augusto H. Ortiz (a1)

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Given an arbitrary associative ring R we consider the ring R[x] of polynomials over R in the commutative indeterminate x. For each radical property S we define the function S * which assigns to each ring R the ideal

of R. It is shown that the property SA (that a ring R be equal to S *(R)) is a radical property. If S is semiprime, then SA is semiprime also. If S is a special radical, then SA is a special radical. SA is always contained in S. A necessary and sufficient condition that S and SA coincide is given.

The results are generalized in the last section to include extensions of R other than R[x], One such extension is the semigroup ring R[A], where A is a semigroup with an identity adjoined.

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References

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1. Amitsur, S. A., A general theory of radicals. II, Amer. J. Math. 76 (1954), 100125.
2. Amitsur, S. A., Radicals of polynomial rings, Can. J. Math. 8 (1956), 355361.
3. Divinsky, N. J., Rings and radicals (Univ. Toronto Press, Toronto, Ontario, 1965).
4. McCoy, N. H., The prime radical of a polynomial ring, Publ. Math. Debrecen 4 (1956), 161162.
5. McCoy, N. H., Certain classes of ideals in polynomial rings, Can. J. Math. 9 (1957), 352362.
6. Nagata, M., On the theory of radicals in a ring, J. Math. Soc. Japan 3 (1951), 330344.
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A Construction in General Radical Theory

  • Augusto H. Ortiz (a1)

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