On normal radicals, N-radicals, and A-radicals

Jaegermann, Michał; Sands, A. D.

doi:10.1016/0021-8693(78)90159-x

Cited by 26 publications

(23 citation statements)

References 12 publications

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“…The aim of this note is to prove that this upper radical is also an TV-radical. Note that the first example of an TV-radical which is not special was found by Jaegermann and Sands [3]All rings in this paper are associative, not necessarily with 1. To denote that / is an ideal (prime ideal, left ideal) of a ring R we write I<R (I<'R, I < R).…”

mentioning

confidence: 57%

A note on prime essential rings

Zand

1994

Bull. Austral. Math. Soc.

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It is proved that the upper radical determined by the class of prime essential rings is an iV-radical.Several examples are known of hereditary supernilpotent radicals which are not special. Recently Gardner and Stewart [2] showed how one can construct many such examples (including most of those already known) using prime essential rings, which were introduced for other reasons by Rowen in [5]. In particular Gardner and Stewart showed that the upper radical determined by the class of these rings is not special. The aim of this note is to prove that this upper radical is also an TV-radical. Note that the first example of an TV-radical which is not special was found by Jaegermann and Sands A ring R is called prime essential [5] if it is semiprime and all its prime ideals are essential.Recall that a radical is an TV-radical [6] if it is left strong (that is, semisimple rings contain no non-zero radical left ideals), left hereditary (that is, left ideals of radical rings are radical) and contains the prime radical 0. Throughout the paper instead of "radical class" we write briefly "radical".All undefined terms and facts used can be found in [7]. We start with some lemmas.(i) Every ideal of a prime essential ring is a prime essential ring.(ii) If I is an essential ideal of R and I is a prime esssential ring then R is a prime essential ring.

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confidence: 57%

A note on prime essential rings

Zand

1994

Bull. Austral. Math. Soc.

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“…Verifications of the appropriate closure properties which are not straight-forward can be found in [1], [2], [3] for semilocal and semiprimary rings, in [6] for nilpotent rings, and in [11], [17] for Pi-rings. The class of Jacobson rings is a left and right hereditary radical class [9], [18] from which fact analogues of properties (ii), (iv), and (v) are immediate. The other properties of Jacobson rings are verified in [2], [4], [16].…”

Section: Iv) a Homomorphic Image Of A Perfect Ring Is Perfect (V) Lementioning

confidence: 99%

“…We are grateful to the referee for drawing our attention to [9]. This enabled us to include the class of Jacobson rings in Proposition 3, replacing a weaker result that we had initially proved for Jacobson rings.…”

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confidence: 99%

Semigroup-graded rings with finite support

1996

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“…In particular we investigate hereditariness of lower strong and stable radicals. Some facts obtained are related to some results and questions of [2,6,7].All rings in the paper are associative. Fundamental definitions and properties of radicals may be found in [9].…”

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confidence: 99%

“…In particular we investigate hereditariness of lower strong and stable radicals. Some facts obtained are related to some results and questions of [2,6,7].…”

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confidence: 99%

Hereditariness of strong and stable radicals

Puczyłowski

1982

Glasgow Math. J.

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The aim of this paper is to discuss some relations among hereditary, strong and stable radicals. In particular we investigate hereditariness of lower strong and stable radicals. Some facts obtained are related to some results and questions of [2,6,7].All rings in the paper are associative. Fundamental definitions and properties of radicals may be found in [9]. Definitions of hereditary and strong radicals are used as in Sands [7]. We say that a radical S is left {right) stable if (p):for every ring R and every left (right) ideal / of R it follows S(I)QS(R).Of course any left (right) stable radical is left (right) strong. If P is a subring of a ring R then P* will denote the ideal of R generated by P. Of course P* = JR 1 PR 1 where R 1 is the natural extension of R to the ring with unity. LEMMA 1. Let S be a radical and P a subring of a ring R such that P°eS. Then (P*)°eS. In particular if S is left strong (stable) then so is S°.Proof. Since P* = R

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On normal radicals, N-radicals, and A-radicals

Cited by 26 publications

References 12 publications

A note on prime essential rings

A note on prime essential rings

Semigroup-graded rings with finite support

Hereditariness of strong and stable radicals

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