On a hereditary radical property relating to the reducedness.

*(English)*Zbl 1219.16009All rings in this paper are associative with identity unless stated otherwise. A ring \(R\) is right (left) weakly regular if \(I^2=I\) for every right (left) ideal of \(R\). A ring is weakly regular if it is both right and left weakly regular. For a ring \(R\), the prime radical and the set of all nilpotent elements are denoted by \(P(R)\) and \(N(R)\), respectively. A ring \(R\) is 2-primal if \(P(R)=N(R)\). A ring \(R\) is called strongly 2-primal if every homomorphic image of \(R\) is 2-primal. For a ring \(R\) and a proper ideal \(I\) of \(R\), \(I\) is called completely prime (respectively, completely semiprime) if \(R/I\) is a domain (respectively, a reduced ring). A ring (possibly without identity) is called reduced if it has no nonzero nilpotent elements. A ring \(R\) (possibly without identity) is called homomorphically reduced if every homomorphic image of \(R\) is reduced.

In this paper the authors study homomorphically reduced rings. In the first section of the paper, the authors examine the relations between homomorphically reduced rings and related concepts. For example, they show that there exist homomorphically reduced rings which are neither left nor right regular and there are many simple rings but not homomorphically reduced. They observe that the classes of homomorphically reduced rings and domains do not contain each other and they give necessary and sufficient conditions for a ring (possibly without identity) to be homomorphically reduced.

A ring \(R\) is (von Neumann) regular if for each \(a\in R\) there exists \(x\in R\) such that \(a=axa\). A ring is called Abelian if every idempotent is central. A ring is right (respectively, left) duo if every right (respectively, left) ideal is two-sided. A ring is called duo if it is both right and left duo. The authors observe that the homomorphical reducedness is equivalent to the regularity and the weak regularity for commutative rings but regular rings need not be homomorphically reduced and homomorphically reduced rings also need not be regular. The authors show that for a regular ring \(R\), the following conditions are equivalent: (1) \(R\) is reduced; (2) \(R\) is Abelian; (3) \(R\) is homomorphically reduced; (4) \(R\) is (strongly) 2-primal; (5) \(R\) is right (left) duo.

A ring \(R\) is strongly regular if \(a\in a^2R\) for each \(a\in R\). The authors show that for a right duo ring \(R\), the following conditions are equivalent: (1) \(R\) is homomorphically reduced; (2) \(R\) is right (left) weakly regular; (3) \(R\) is strongly regular (if and only if \(R\) is regular); Every homomorphic image of \(R\) is right nonsingular. Consequently, they obtain that \(R\) is homomorphically reduced if and only if \(R\) is weakly regular if and only if \(R\) is regular when \(R\) is a commutative ring.

A ring \(R\) is strongly prime if \(R\) is prime with no nonzero nil ideals and an ideal \(P\) of \(R\) is called strongly prime if \(R/P\) is strongly prime. The authors show that for a right (or left) duo reduced ring \(R\), \(R\) is homomorphically reduced if and only if \(R\) is weakly regular if and only if \(R\) is (strongly) regular if and only if every (strongly) prime ideal of \(R\) is maximal if and only if every (strongly) prime factor ring of \(R\) is a simple domain if and only if every (strongly) prime factor ring of \(R\) is a division ring. They conclude that for a commutative reduced ring \(R\), \(R\) is homomorphically reduced if and only if every (strongly) prime ideal of \(R\) is maximal if and only if every (strongly) prime factor ring of \(R\) is a field. The authors show that the center of a homomorphically reduced ring is homomorphically reduced. Moreover, the center of a homomorphically reduced ring is commutative regular and a homomorphically reduced ring is indecomposable (as a ring) if and only if its center is a field. This allows them to conclude that a commutative domain \(R\) is homomorphically reduced if and only if \(R\) is a field if and only if \(R\) is self-injective.

A proper ideal of a ring is called homomorphically reduced if it is homomorphically reduced as a ring. In the second section of the paper, the authors show that homomorphically reducedness is a hereditary radical property and \(Hr(R)=\{x\in R:RxR\) is a homomorphically reduced ideal of \(R\}\) is the ideal of a ring \(R\) maximal with respect to the homomorphical reducedness. They call this ideal the hr-radical of \(R\). The authors call a ring \(R\) hr-semisimple if \(Hr(R)=0\). The authors show that the ring \(\mathbb Z\) of all integers is hr-semisimple and so are the \(n\times n\) full matrix ring \(\text{Mat}_n(R)\) (\(n\geq 2\)) over \(R\), the polynomial ring \(R[X]\) and the power series ring \(R[[X]]\) with an indeterminate \(X\) (possibly infinite) over \(R\) for any ring \(R\). Also, the authors study the dual concept of the radical \(Hr(-)\). They show that any ring \(R\) (possibly without unity) has an ideal \(S\) such that (1) \(S/K\) is not homomorphically reduced for each proper ideal \(K\) of \(S\) and (2) If \(L\) is an ideal of \(R\) with \(L\varsubsetneqq S\), then \(L/H\) is homomorphically reduced for some ideal \(H\) of \(R\) with \(H\subsetneqq L\). They also show that if \(R\) is a homomorphically reduced ring and \(I\) is a proper ideal of \(R\), then the following conditions are equivalent: (1) \(I\) is a prime ideal of \(R\); (2) \(I\) is a completely prime ideal of \(R\); (3) \(R\setminus I\) is a multiplicative semigroup of \(R\setminus 0\). Moreover, they give a condition for homomorphically reduced rings to be weakly regular and vice versa.

In the third section of the paper, the authors examine and construct more examples of homomorphically reduced rings. A ring \(R\) is called right Ore if given \(a,b\in R\) with \(b\) regular, there exist \(a_1,b_1\in R\) with \(b_1\) regular such that \(ab_1=ba_1\). The authors show that if \(R\) is a right Ore ring and \(Q\) is the classical right quotient ring of \(R\), then if \(R\) is homomorphically reduced, then so is \(Q\) but the converse need not hold. They show that if \(R\) is a PID, then \(R/(p_1\cdots p_k)\) is homomorphically reduced for distinct primes \(p_1,\dots,p_k\) in \(R\). The authors show that the direct product \(\prod_{i\in I}R_i\) (respectively, the direct sum \(\bigoplus _{i\in I}R_i\)) of rings \(R_i\) (\(i\in I\)) is homomorphically reduced if and only if so is every \(R_i\).

Let \(R\) be an algebra over a commutative ring \(S\). The Dorroh extension \(R\oplus_DS\) of \(R\) by \(S\) is the ring with operations \((r_1,s_1)+(r_2,s_2)=(r_1+r_2,s_1+s_2)\) and \((r_1,s_1)(r_2,s_2)=(r_1r_2+s_1r_2+s_2r_1,s_1s_2)\), where \(r_i\in R\) and \(s_i\in S\). The authors prove that if \(R\) is a simple domain (possibly without identity) that is an algebra over a field \(F\) satisfying the condition that \(a+\alpha\neq 0\) when \(0\neq a\in R\) and \(0\neq\alpha\in F\), then \(R\oplus_DF\) is homomorphically reduced. They also show that the class of homomorphically reduced rings is closed under direct limits and finite subdirect products. However, subdirect products of infinitely many homomorphically reduced rings need not be homomorphically reduced. Moreover, the authors show that if \(R\) is a homomorphically reduced ring, then any homomorphic image \(S\) of \(R\) is left self-injective if and only if \(S\) is left self-injective.

For a ring \(R\), the classical right quotient ring of \(R\) (if exists) and the maximal right quotient ring of \(R\) are denoted by \(Q_{cl}^r(R)\) and \(Q_{\max}^r(R)\), respectively. The authors show that if \(R\) is a reduced right Ore ring with the ACC (or DCC) for annihilator ideals of \(R\), then \(Q_{\max}^r(R)\) (\(=Q_{cl}^r(R)\)) is strongly regular (hence homomorphically reduced). They note that the condition “right Ore” is not superfluous.

The uniform dimension of a module \(M\) is denoted by \(u.\dim M\). The authors close the paper by showing that if \(R\) is a reduced right Ore ring with the ACC (or DCC) for annihilator ideals of \(R\) and \(u.\dim R_R\) is finite, then \(Q_{\max}^r(R)\) (\(=Q_{cl}^r(R)\)) is a finite direct product of division rings.

In this paper the authors study homomorphically reduced rings. In the first section of the paper, the authors examine the relations between homomorphically reduced rings and related concepts. For example, they show that there exist homomorphically reduced rings which are neither left nor right regular and there are many simple rings but not homomorphically reduced. They observe that the classes of homomorphically reduced rings and domains do not contain each other and they give necessary and sufficient conditions for a ring (possibly without identity) to be homomorphically reduced.

A ring \(R\) is (von Neumann) regular if for each \(a\in R\) there exists \(x\in R\) such that \(a=axa\). A ring is called Abelian if every idempotent is central. A ring is right (respectively, left) duo if every right (respectively, left) ideal is two-sided. A ring is called duo if it is both right and left duo. The authors observe that the homomorphical reducedness is equivalent to the regularity and the weak regularity for commutative rings but regular rings need not be homomorphically reduced and homomorphically reduced rings also need not be regular. The authors show that for a regular ring \(R\), the following conditions are equivalent: (1) \(R\) is reduced; (2) \(R\) is Abelian; (3) \(R\) is homomorphically reduced; (4) \(R\) is (strongly) 2-primal; (5) \(R\) is right (left) duo.

A ring \(R\) is strongly regular if \(a\in a^2R\) for each \(a\in R\). The authors show that for a right duo ring \(R\), the following conditions are equivalent: (1) \(R\) is homomorphically reduced; (2) \(R\) is right (left) weakly regular; (3) \(R\) is strongly regular (if and only if \(R\) is regular); Every homomorphic image of \(R\) is right nonsingular. Consequently, they obtain that \(R\) is homomorphically reduced if and only if \(R\) is weakly regular if and only if \(R\) is regular when \(R\) is a commutative ring.

A ring \(R\) is strongly prime if \(R\) is prime with no nonzero nil ideals and an ideal \(P\) of \(R\) is called strongly prime if \(R/P\) is strongly prime. The authors show that for a right (or left) duo reduced ring \(R\), \(R\) is homomorphically reduced if and only if \(R\) is weakly regular if and only if \(R\) is (strongly) regular if and only if every (strongly) prime ideal of \(R\) is maximal if and only if every (strongly) prime factor ring of \(R\) is a simple domain if and only if every (strongly) prime factor ring of \(R\) is a division ring. They conclude that for a commutative reduced ring \(R\), \(R\) is homomorphically reduced if and only if every (strongly) prime ideal of \(R\) is maximal if and only if every (strongly) prime factor ring of \(R\) is a field. The authors show that the center of a homomorphically reduced ring is homomorphically reduced. Moreover, the center of a homomorphically reduced ring is commutative regular and a homomorphically reduced ring is indecomposable (as a ring) if and only if its center is a field. This allows them to conclude that a commutative domain \(R\) is homomorphically reduced if and only if \(R\) is a field if and only if \(R\) is self-injective.

A proper ideal of a ring is called homomorphically reduced if it is homomorphically reduced as a ring. In the second section of the paper, the authors show that homomorphically reducedness is a hereditary radical property and \(Hr(R)=\{x\in R:RxR\) is a homomorphically reduced ideal of \(R\}\) is the ideal of a ring \(R\) maximal with respect to the homomorphical reducedness. They call this ideal the hr-radical of \(R\). The authors call a ring \(R\) hr-semisimple if \(Hr(R)=0\). The authors show that the ring \(\mathbb Z\) of all integers is hr-semisimple and so are the \(n\times n\) full matrix ring \(\text{Mat}_n(R)\) (\(n\geq 2\)) over \(R\), the polynomial ring \(R[X]\) and the power series ring \(R[[X]]\) with an indeterminate \(X\) (possibly infinite) over \(R\) for any ring \(R\). Also, the authors study the dual concept of the radical \(Hr(-)\). They show that any ring \(R\) (possibly without unity) has an ideal \(S\) such that (1) \(S/K\) is not homomorphically reduced for each proper ideal \(K\) of \(S\) and (2) If \(L\) is an ideal of \(R\) with \(L\varsubsetneqq S\), then \(L/H\) is homomorphically reduced for some ideal \(H\) of \(R\) with \(H\subsetneqq L\). They also show that if \(R\) is a homomorphically reduced ring and \(I\) is a proper ideal of \(R\), then the following conditions are equivalent: (1) \(I\) is a prime ideal of \(R\); (2) \(I\) is a completely prime ideal of \(R\); (3) \(R\setminus I\) is a multiplicative semigroup of \(R\setminus 0\). Moreover, they give a condition for homomorphically reduced rings to be weakly regular and vice versa.

In the third section of the paper, the authors examine and construct more examples of homomorphically reduced rings. A ring \(R\) is called right Ore if given \(a,b\in R\) with \(b\) regular, there exist \(a_1,b_1\in R\) with \(b_1\) regular such that \(ab_1=ba_1\). The authors show that if \(R\) is a right Ore ring and \(Q\) is the classical right quotient ring of \(R\), then if \(R\) is homomorphically reduced, then so is \(Q\) but the converse need not hold. They show that if \(R\) is a PID, then \(R/(p_1\cdots p_k)\) is homomorphically reduced for distinct primes \(p_1,\dots,p_k\) in \(R\). The authors show that the direct product \(\prod_{i\in I}R_i\) (respectively, the direct sum \(\bigoplus _{i\in I}R_i\)) of rings \(R_i\) (\(i\in I\)) is homomorphically reduced if and only if so is every \(R_i\).

Let \(R\) be an algebra over a commutative ring \(S\). The Dorroh extension \(R\oplus_DS\) of \(R\) by \(S\) is the ring with operations \((r_1,s_1)+(r_2,s_2)=(r_1+r_2,s_1+s_2)\) and \((r_1,s_1)(r_2,s_2)=(r_1r_2+s_1r_2+s_2r_1,s_1s_2)\), where \(r_i\in R\) and \(s_i\in S\). The authors prove that if \(R\) is a simple domain (possibly without identity) that is an algebra over a field \(F\) satisfying the condition that \(a+\alpha\neq 0\) when \(0\neq a\in R\) and \(0\neq\alpha\in F\), then \(R\oplus_DF\) is homomorphically reduced. They also show that the class of homomorphically reduced rings is closed under direct limits and finite subdirect products. However, subdirect products of infinitely many homomorphically reduced rings need not be homomorphically reduced. Moreover, the authors show that if \(R\) is a homomorphically reduced ring, then any homomorphic image \(S\) of \(R\) is left self-injective if and only if \(S\) is left self-injective.

For a ring \(R\), the classical right quotient ring of \(R\) (if exists) and the maximal right quotient ring of \(R\) are denoted by \(Q_{cl}^r(R)\) and \(Q_{\max}^r(R)\), respectively. The authors show that if \(R\) is a reduced right Ore ring with the ACC (or DCC) for annihilator ideals of \(R\), then \(Q_{\max}^r(R)\) (\(=Q_{cl}^r(R)\)) is strongly regular (hence homomorphically reduced). They note that the condition “right Ore” is not superfluous.

The uniform dimension of a module \(M\) is denoted by \(u.\dim M\). The authors close the paper by showing that if \(R\) is a reduced right Ore ring with the ACC (or DCC) for annihilator ideals of \(R\) and \(u.\dim R_R\) is finite, then \(Q_{\max}^r(R)\) (\(=Q_{cl}^r(R)\)) is a finite direct product of division rings.

Reviewer: Halina France-Jackson (Port Elizabeth)

##### MSC:

16D80 | Other classes of modules and ideals in associative algebras |

16N80 | General radicals and associative rings |

16D25 | Ideals in associative algebras |

16E50 | von Neumann regular rings and generalizations (associative algebraic aspects) |

16U20 | Ore rings, multiplicative sets, Ore localization |

16N60 | Prime and semiprime associative rings |

16N40 | Nil and nilpotent radicals, sets, ideals, associative rings |

16U80 | Generalizations of commutativity (associative rings and algebras) |

##### Keywords:

hereditary radicals; homomorphically reduced rings; right Ore rings; weakly regular rings; prime radical; nilpotent elements; von Neumann regular rings; Abelian rings; right duo rings; strongly prime ideals; subdirect products
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\textit{C. M. Ha} et al., Commun. Algebra 39, No. 2, 608--620 (2011; Zbl 1219.16009)

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