Ring properties of amalgamated products are investigated. We offer new, elementary arguments which extend results from [5] and [12] to noncommutative setting and also give new properties of amalgamated rings

Nowakowska, Marta

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Annales Mathematicae Silesianae 35 (2021), no. 2, 282–288DOI: 10.2478/amsil-2021-0010A NOTE ON AMALGAMATED RINGS ALONG AN IDEALMarta NowakowskaAbstract. Ring properties of amalgamated products are investigated. We of-fer new, elementary arguments which extend results from [5] and [12] to non-commutative setting and also give new properties of amalgamated rings.1. Introduction and preliminariesAll rings in this paper are associative, we do not assume they containunity.We write I CA, if I is an ideal of a ring A. We say that an ideal I of A isessential, if I ∩ J 6= {0} for every {0} 6= J CA.Let R be a class of rings. Then:(a) R is closed under extensions, if the following implication holds:I CA, I ∈ R and A/I ∈ R =⇒ A ∈ R.(b) R is homomorphically closed, if every homomorphic image of a ring fromR is in R.(c) R is closed under subrings, if every subring of a ring belonging to R isin R.(d) R is hereditary, if I CA ∈ R implies that I ∈ R.Received: 05.05.2021. Accepted: 28.07.2021. Published online: 30.08.2021.(2020) Mathematics Subject Classification: 16S99, 16N20, 16N40, 16N60, 16P20,16P40.Key words and phrases: amalgamated ring, radical, prime ring, reduced ring, Noether-ian ring.c©2021 The Author(s).This is an Open Access article distributed under the terms of the Creative Commons Attribution LicenseCC BY (http://creativecommons.org/licenses/by/4.0/).A note on amalgamated rings along an ideal 283A class of rings γ is called a radical class or shortly a radical (cf. [10]), if itis closed under extensions, homomorphically closed and∑{ICA | I ∈ γ} ∈ γfor every ring A.For a given ring A and a non-empty subset X of A we denote by lA(X) ={a ∈ A | aX = {0}} the left annihilator of X.We write R = S⊕. I, if R = S+ I, S is a subring of a ring R, I is an idealof R and S ∩ I = {0}. Clearly S ⊕. I/I ∼= S.Let A,B be associative rings, J be an ideal of B and f : A −→ B a ringhomomorphism (we do not assume that f preserves identity even in case ofunital rings). We consider the following subring of A×B:A ./f J = {(a, f(a) + j) | a ∈ A, j ∈ J}which is called the amalgamation of A with B along J with respect to f . Wewill say shortly that A ./f J is the amalgamated ring.The above definition was introduced by M. D’Anna, C.A. Finocchiaro andM. Fontana in case of commutative unital rings (cf. [5]). Various classicalconstructions can be presented as particular cases of an amalgamation, forexample amalgamated duplication of a ring along an ideal (cf. [4, 7]), Nagata’sidealization also called the trivial ring extension (cf. [14]),D+M constructionsor CPI-extensions (cf. [5]). Moreover, the amalgamation A ./f J is relatedto a construction introduced by D.D. Anderson in [1] and motivated by aclassical embedding of a ring without identity into a ring with identity (cf.[8]). Furthermore, there are some applications of amalgamated algebras inalgebraic geometry which follow from the fact that it is possible to iterate theamalgamation of algebras and the result is still an amalgamated algebra (cf.[5]). Moreover an amalgamation can be realized as a pullback of mappingsand some pullbacks give rise to amalgamated rings (cf. [5]).There are systematic studies concerning amalgamated rings, but almost allof them are restricted to commutative rings with unity. The exception is [12],where associative unital rings are considered. For more details see for example[2, 3, 6, 9, 11, 13].The main goal of this paper is to show that using elementary tools of aring theory we can easily obtain generalizations of many results from otherpapers (see Corollary 2.7). In particular, we extend some of results from [5]and [12] to associative rings. They turn out to be direct consequences ofgeneral Theorems 2.4 and 2.5.284 Marta Nowakowska2. Main resultsWe start with some basic observations. The following proposition containsstraightforward properties of amalgamated rings. They correspond to thosepresented in [5, Proposition 5.1].Proposition 2.1. Let A,B be associative rings, J C B and f : A −→ Bbe a ring homomorphism. Then:(i) A ∼= {(a, f(a)) | a ∈ A} is a subring of A ./f J ,(ii) J ∼= {0} × J is an ideal of A ./f J and A ./f J/({0} × J) ∼= A,(iii) f−1(J)× J and f−1(J)× {0} are ideals of A ./f J andA ./f J/(f−1(J)× {0}) ∼= f(A) + J.Remark 2.2.(a) Notice that A ./f J ⊆ A × (f(A) + J) and f : A −→ f(A) + J ⊆ B.Hence, if we restrict B to f(A) + J and denote f̄ : A −→ f(A) + J , thenA ./f J = A ./f̄ J . Therefore, without loss of generality, in the definitionof amalgamated rings we may write f(A) + J instead of B.(b) Note that {(a, f(a)) | a ∈ A}∩ ({0}× J) = {(0, 0)}. Thus Proposition 2.1(i), (ii) yields that A ./f J ∼= A⊕. J .(c) The statement (b) implies that:(i) J = {0} if and only if A ./f J ∼= A,(ii) A = {0} if and only if A ./f J ∼= J .Accordingly to the above remark we will consider the situation whenA 6= {0}, J 6= {0} and B = f(A) + J . Then we say that the amalgamatedring A ./f J is proper.In the following lemma we present how properties of the class R affectproperties of rings A, J and B. They easily follow from Proposition 2.1.Lemma 2.3. Let R be a class of rings and A ./f J a proper amalgamatedring such that A ./f J ∈ R.(i) If R is closed under subrings, then A ∈ R and J ∈ R.(ii) If R is homomorphically closed, then A ∈ R and B ∈ R.(iii) If R is hereditary, then J ∈ R.Let the class R be closed under extensions, A ∈ R and J ∈ R. Thus thestatement (ii) of Proposition 2.1 yields that A ∼= A ./f J/({0} × J) ∈ R.Hence, by assumptions, we conclude that A ./f J ∈ R. Therefore, by Lemma2.3 (i), we prove the following result.A note on amalgamated rings along an ideal 285Theorem 2.4. Assume A ./f J is a proper amalgamated ring and R isthe class of rings closed under extensions and subrings. Then the followingconditions are equivalent:(i) A ./f J ∈ R,(ii) A ∈ R and J ∈ R.The above theorem shows that this type of properties of A ./f J do notdepend on properties of a homomorphism f .Note that, if R is closed under extensions and subrings, A ∈ R and B ∈ R,then A × B ∈ R, so A ./f J ∈ R. However, in a view of Remark 5.5 in [5],the condition (ii) of Theorem 2.4 can not be replaced by A ∈ R and B ∈ R.Below we present what additional property should have the class R to getthe opposite implication in Lemma 2.3 (ii).Theorem 2.5. Assume A ./f J is a proper amalgamated ring and R is theclass of rings closed under extensions, homomorphically closed and hereditary.Then the following conditions are equivalent:(i) A ./f J ∈ R,(ii) A ∈ R and B ∈ R.Proof. The implication (i) =⇒ (ii) follows from Lemma 2.3 (ii).Assume that condition (ii) holds. By Proposition 2.1 we get thatf−1(J) × J/({0} × J) C A ./f J/({0} × J) ∼= A ∈ R. Since the class Ris hereditary, f−1(J)× J/({0}× J) ∼= f−1(J)×{0} ∈ R. Proposition 2.1 (iii)implies that A ./f J/(f−1(J)× {0}) ∼= B ∈ R. Hence, by assumption that Ris closed under extensions, we prove (i). The following conclusion is a direct consequence of Theorem 2.5.Corollary 2.6. Let γ be a hereditary, radical class and A ./f J a properamalgamated ring. Then the following conditions are equivalent:(i) A ./f J ∈ γ,(ii) A ∈ γ and B ∈ γ.The fact below presents some applications of Theorems 2.4 and 2.5.Corollary 2.7.(1) If R is one of the classes of rings listed below:(i) the class of reduced rings,(ii) the class of nilpotent rings,(iii) the class of nil rings.Then, by Theorem 2.4, we get a description of amalgamated rings belong-ing to R.286 Marta Nowakowska(2) If R is one of the classes of rings listed below:(i) the class of Noetherian rings,(ii) the class of Artinian rings,(iii) the class of von Neumann regular rings,(iv) the class of strongly regular rings,(v) the Jacobson radical class,(vi) the prime radical class,(vii) the generalized nil radical class,(viii) the Brown-McCoy radical class,(ix) the Levitzki radical class.Then, by Theorem 2.5, we get a description of amalgamated rings belong-ing to R.Some of the classes presented above were considered in [3, 5] and [12].Hence we obtain generalizations of Propositions: 5.6 in [5], 2.1 in [12] and 2.21in [3]. Moreover our arguments are different and much simpler than those in[3, 5] and [12].Let I := f−1(J) × {0} and K := {0} × J . Then I · K = {(0, 0)}. Thisequality will turn out to be crucial in a description of primness of amalgamatedrings.The following result generalizes Theorems 2.3 and 2.4 in [12] and Proposi-tion 5.2 in [5]. Our proof is based on the above observation, so it is completelydifferent and much shorter than those in [12] and [5].Theorem 2.8. Assume A ./f J is a proper amalgamated ring. Then thefollowing conditions are equivalent:(i) A ./f J is a prime ring (a domain).(ii) B is a prime ring (a domain) and f−1(J) = {0}.Proof. (i) =⇒ (ii) Assume A ./f J is a prime ring. By Proposition 2.1(ii), (iii), I := f−1(J) × {0} C A ./f J and K := {0} × J C A ./f J . SinceI ·K = {(0, 0)}, so the primness of the ring A ./f J implies that I = {(0, 0)}or K = {(0, 0)}.If K = {(0, 0)}, then J = {0}, a contradiction. Therefore I = {0}, whichyields that f−1(J) = {0}. Hence A ./f J ∼= B, by the statement (iii) ofProposition 2.1. Thus B is a prime ring.If A ./f J is a domain, then clearly it is a prime ring. Hence applying theabove argumentation we show that B is a domain and f−1(J) = {0}.The implication (ii) =⇒ (i) follows directly from Proposition 2.1 (iii). A note on amalgamated rings along an ideal 287Below we collect well-known facts concerning prime rings and domains.Lemma 2.9. Let {0} 6= I CR.(1) The following conditions are equivalent.(i) R is a prime ring.(ii) I is a prime ring and I is an essential ideal of R.(iii) I is prime ring and lR(I) = {0}.(2) If I is an essential ideal of R and I is a domain, then R is a domain.By the above lemma and Theorem 2.8 we obtain the following two results.Corollary 2.10. Assume A ./f J is a proper amalgamated ring. Thenthe following conditions are equivalent:(i) A ./f J is a prime ring.(ii) J is a prime ring, lB(J) = {0} and f−1(J) = {0}.(iii) J is a prime ring, J is an essential ideal of B and f−1(J) = {0}.Corollary 2.11. Assume A ./f J is a proper amalgamated ring. Thenthe following conditions are equivalent:(i) A ./f J is a domain.(ii) J is a domain, lB(J) = {0} and f−1(J) = {0}.Proof. Assume (i) holds. Then J is a domain and A ./f J is a primering. Hence, by Corollary 2.10, we get (ii).Suppose (ii) is true. Since J is a domain, it is also a prime ring. ThusCorollary 2.10 implies that J is an essential ideal of B. Therefore, by thestatement (2) of Corollary 2.10, we conclude that B is a domain. Finally,applying Theorem 2.8 we obtain (i).Acknowledgments. The autor would like to thank Professor JerzyMatczuk for many helpful comments and suggestions which improved thepaper in a significant way.References[1] D.D. Anderson, Commutative rings, in: J.W. Brewer et al. (Eds.), Multiplicative IdealTheory in Commutative Algebra. A Tribute to the Work of Robert Gilmer, Springer,New York, 2006, pp. 1–20.[2] M. Chhiti and N. Mahdou, Some homological properties of amalgamated duplicationof a ring along an ideal, Bull. Iranian Math. Soc. 38 (2012), no. 2, 507–515.288 Marta Nowakowska[3] M. Chhiti, N. Mahdou, and M. Tamekkante,Clean property in amalgamated algebrasalong an ideal, Hacet. J. Math. Stat. 44 (2015), no. 1, 41–49.[4] M. D’Anna, A construction of Gorenstein rings, J. Algebra 306 (2006), no. 2, 507–519.[5] M. D’Anna, C.A. Finocchiaro, and M. Fontana, Amalgamated algebras along an ideal,in: M. Fontana et al. (Eds.), Commutative Algebra and its Applications, Walter deGruyter, Berlin, 2009, pp. 155–172.[6] M. D’Anna, C.A. Finocchiaro, and M. Fontana, Properties of chains of prime idealsin amalgamated algebra along an ideal, J. Pure Appl. Algebra 214 (2010), no. 9,1633–1641.[7] M. D’Anna and M. Fontana, An amalgamated duplication of a ring along an ideal: thebasic propeties, J. Algebra Appl. 6 (2007), no. 3, 443–459.[8] J.L. Dorroh, Concerning adjunctions to algebras, Bull. Amer. Math. Soc. 38 (1932),no. 2, 85–88.[9] M. El Maalmi and H. Mouanis, On Steinitz-like properties in amalgamated algebrasalong an ideal, J. Algebra Appl. 19 (2020), no. 12, 2050237, 9 pp.[10] B.J. Gardner and R. Wiegandt, Radical Theory of Rings, Monographs and Textbooksin Pure and Applied Mathematics, 261, Marcel Dekker, Inc., New York, 2004.[11] S. Kabbaj, K. Louariti, and M. Tamekkante, Bi-amalgamated algebras along ideals,J. Commut. Algebra 9 (2017), no. 1, 65–87.[12] H. Kose, Y. Kurtulmaz, B. Ungor, and A. Harmanci, Semicommutativity of amalga-mated rings, J. Math. Res. Appl. 38 (2018), no. 4, 366–376.[13] A. Mimouni, Clean-like properties in pullbacks and amalgamation rings, Acta Math.Hungar. 156 (2018), no. 1, 91–101.[14] M. Nagata, Local Rings, Interscience Tracts in Pure and Applied Mathematics, 13,Interscience Publishers a division of John Wiley & Sons, New York, 1962.Institute of MathematicsUniversity of Silesia in KatowiceBankowa 1440-007 KatowicePolande-mail: marta.nowakowska@us.edu.pl

A note on amalgamated rings along an ideal

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A note on amalgamated rings along an ideal

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