Algebraic Properties of Semi-Direct Sums of Rings

A fundamental problem in Ring Theory is to determine or describe ideal extensions of a given ring A by another ring B. More precisely the problem can be formulated as follows: for given rings A, B describe all rings R such that A ◁ R and R/A ≅ B. An equally important problem concerns a description of all ideal extensions of a given ring. Many authors have considered this problem in various contexts e.g., C.J. Everett in [16] presented an axiomatic description of ideal extensions of a given ring. His research was continued by S. Mac Lane in [21] who used a homological treatment. Another approach to this problem was provided by M. Petrich in [27] and R.M. Raphael and W.D. Burgess in [6].

If R = S + I, where S is a subring of a ring R, I is an ideal of R and S ∩ I = {0}, then a ring R is said to be a semi-direct sum of rings S and I and we write R = S ⊕_▷ I. A semi-direct sum is an example of an ideal extension. Moreover, it is a natural generalization of a direct sum of rings and corner rings (cf. [22]). Semi-direct sums of rings are also called general corner rings or general corner extensions (cf. [20]) and generalize the notion of Pierce decomposition. Furthermore, semi-direct sums are examples of Schreier extensions (cf. [28]).

In [11] M. D’Anna, C.A. Finocchiaro and M. Fontana introduced the notion of amalgamated rings in case of commutative unital rings (cf. [11]). Various classical constructions can be presented as particular cases of an amalgamation, for example amalgamated duplication of a ring along an ideal (cf. [10], [13]), Nagata’s idealization also called the trivial ring extension (cf. [24]), D + M constructions or CPI-extensions (cf. [11]). Furthermore, the amalgamation is related to a construction introduced by D.D. Anderson in [2] and motivated by a classical embedding of a ring without unity into a ring with unity (cf. [14]). Moreover, there are some applications of amalgamated algebras in algebraic geometry which follow from the fact that it is possible to iterate the amalgamation of algebras and the result is still an amalgamated algebra (cf. [11]). Additionally, an amalgamation can be realized as a pullback of mappings, and some pullbacks give rise to amalgamated rings (cf. [11]).

There are systematic studies concerning amalgamated rings, but most of them are restricted to commutative rings with unity. For more details, see for example [7], [8], [12], [15], [19], [23], [26].

We show that the class of semi-direct sums of rings and the class of amalgamated rings coincide. This is done in Section 3. This allows us to obtain characterizations of semi-direct sums which are closed under some algebraic constructions. Moreover, we improve some results from [26].

In Section 4, we provide a description of the Jacobson radical of a semidirect sum of rings and amalgamated rings, applying [28, Theorem 3]. We show when the Jacobson radical of a semi-direct sum of rings A and J is a semi-direct sum of the Jacobson radicals of A and J.

The main theorem of Section 5 is a characterization of semi-direct sums of rings which are left Steinitz. This result extends [15, Theorem 2.1] to associative rings. We present a short and elementary proof of that fact using description of the Jacobson radical from Section 4. We also show when a semi-direct sum of rings is local and T-nilpotent.

All rings in this paper are associative but not necceasarily with unity.

We write I ◁ R, if I is an ideal of a ring R. We say that an ideal I of R is essential, if I ∩ J ≠ {0} for every {0} ≠ J ◁ R.

If R = S ⊕_▷ I is a direct-sum of S and I, then clearly S ⊕_▷ I/I ≅ S. We will consider only the situation when S ≠ {0} ≠ I. Moreover, if S ◁ R, then R = S ⊕ I, where ⊕ denotes a direct sum of rings S and I.

For a given ring R and a non-empty subset X of R we denote by l_R(X) = {r ∈ R | aX = {0}} the left annihilator of X. If L is a left ideal of R, then clearly l_R(L) ◁ R.

A ring R is called prime, if for every I, J ◁ R, from the equality IJ = {0} follows that either I = {0} or J = {0}.

An ideal I of a ring R is called semiprime, if for every left ideal L of R, whenever L² ⊆ I, then L ⊆ I.

Let ℛ be a class of rings. Then:

• (a)

  ℛ is closed under extensions, if the following implication holds: I⊲A,I∈R and A/I∈R⟹A∈R. I \triangleleft A,I \in {\cal R}\;\;\;{\rm{and}}\;\;\;A/I \in {\cal R} \Longrightarrow A \in {\cal R}.

• (b)

  ℛ is homomorphically closed, if every homomorphic image of a ring from ℛ is in ℛ.

• (c)

  ℛ is closed under subrings, if every subring of a ring belonging to ℛ is in ℛ.

• (d)

  ℛ is hereditary, if I ◁ A ∈ ℛ implies that I ∈ ℛ.

A class of rings γ is called a radical class or shortly a radical (cf. [18]), if it is closed under extensions, homomorphically closed and γ(A) := ∑{I ◁ A | I ∈ γ} ∈ γ for every ring A. If A ∈ γ, we say that A is γ-radical, i.e. γ(A) = A.

An element r ∈ R is called quasi-regular, if there exists s ∈ R such that r+s−rs = 0. An ideal I of a ring R is called quasi-regular, if its every element is quasi-regular.

By 𝒥 (R) we denote the Jacobson radical of a ring R. It is well known that 𝒥 (R) is a semiprime ideal and the largest quasi-regular ideal of R. Furthermore, the following equalities hold JR=r∈R | Rr is quasi-regular=r∈R | rR is quasi-regular=r∈R | rR⊆JR=r∈R | Rr⊆JR. \matrix{ {{\cal J}\left( R \right)} \hfill & { = \left\{ {r \in R\;|\;Rr\;{\rm{is}}\;{\text{quasi-regular}}} \right\} = \left\{ {r \in R\;|\;\;rR\;{\rm{is}}\;{\text{quasi-regular}}} \right\}} \hfill \cr {} \hfill & { = \left\{ {r \in R\;|\;rR \subseteq {\cal J}\left( R \right)} \right\} = \left\{ {r \in R\;|\;Rr \subseteq {\cal J}\left( R \right)} \right\}.} \hfill \cr }

Clearly, if R is a 𝒥-radical, then 𝒥 (R) = R.

Recall that R is a local ring, if the quotient ring R/𝒥 (R) is a division ring.

A ring R is left T-nilpotent, if for every sequence (a_i)_i∈ℕ of elements of R there is an integer n such that a₁a₂ · . . . · a_n = 0. An ideal of a ring is called left T-nilpotent, if it is left T-nilpotent as a ring. [17, Theorem 1.1] shows that the class of T-nilpotent rings is closed under subrings, homomorphic images, extensions and direct sums.

A ring R is said to be a left Steinitz ring (cf. [9]), if every linearly independent subset of a finitely generated free left R-module F can be extended to a basis of F by adjoining elements of a given basis of F. Analogously we define a right Steinitz ring. If a ring is both left and right Steinitz, then we call a such ring a Steinitz ring. B.S. Chwe and J. Neggers in [9] proved that a ring is left Steinitz if and only if it is local and its Jacobson radical is left T-nilpotent.

Recall that a unital ring R is called (uniquely) clean, if every element of R can be written (uniquely) as the sum of a unit and an idempotent. The concept of clean rings was introduced by W.K. Nicholson in [24].

Let A,B be associative rings, J be an ideal of B and f : A ⟶ B a ring homomorphism (we do not assume that f preserves identity even in case of unital rings). We consider the following subring of A × B: A ⋈fJ=a,fa+j | a∈A,j∈J A \;{\bowtie ^f}J = \left\{ {\left( {a,\;f\left( a \right) + j} \right)\;|\;a \in A,\;j \in J} \right\} which is called the amalgamation of A with B along J with respect to f. We will say shortly that A ⋈^f J is the amalgamated ring.

Throughout the paper, following [26, Remark 2.2], we assume that B=fA+J,A≠0 and J≠0. B = f\left( A \right) + J,A \ne \left\{ 0 \right\}\;\;\;{\rm{and}}\;\;\;J \ne \left\{ 0 \right\}.

Moreover, by [26, Proposition 2.1] we know that A ≅ {(a, f(a)) | a ∈ A} is a subring of A ⋈^f J, J ≅ {0} × J is an ideal of A ⋈^f J and (A ⋈^f J)/J ≅ A.

We start this section with a observation that follows directly from [26, Remark 2.2]. Namely from the forementioned remark and under the above notation we know that (3.1) A⊕⊳J≅A ⋈fJ. A\;{ \oplus _ \triangleright }\;J \cong A \;{\bowtie ^f}J. Furthermore (3.2) A⊕⊳J≅A ⋈idJ, A\;{ \oplus _ \triangleright }\;J \cong A \;{\bowtie ^{id}}J, since a mapping φ: A ⊕_▷ J ⟶ A ⋈^id J given by φa+j=a,a+j \varphi \left( {\left( {a + j} \right)} \right) = \left( {a,\;a + j} \right) is a ring isomorphism. Therefore we immediately get the following corollary.

Theorems 3.2 and 3.3 and Corollary 3.4 present characterizations of semidirect sums belonging to classes of rings closed under some constructions.

It is straightforward to see that [26, Theorem 2.4] and (3.1) imply the following result.

Theorem 3.3 and Corollary 3.4 are new versions of [26, Theorem 2.5] and [26, Corollary 2.6], respectively.

Corollary 3.4 directly follows from Theorem 3.3.

Next result gives necessary and sufficient conditions for a semi-direct sum to be a prime ring and a domain. It follows from well-known properies of prime rings, but we include the proof for completeness. We will use Andrunakievich’s Lemma (see [4]) which says that, if J ◁ K ◁ R, then J̄³ ⊆ J, where J̄ denotes an ideal of R generated by J.

In this section we focus on the Jacobson radical of a semi-direct sum of rings.

Since semi-direct sums and amalgamated rings are Schreier extensions, so immediately from [28, Theorem 3] we get the following description of their Jacobson radical.

The following fact will play an important role in the rest of the paper.

The following result for amalgamated rings can be proven analogously as Proposition 4.2, Proposition 4.2(iii) and Proposition 4.3 for semi-direct sums applying Proposition 4.1(ii).

Corollary 4.4.

Let A ⋈^f J be an amalgamated ring.

• (i)

  The following conditions are equivalent:

  • (1)

    f(𝒥 (A))J ⊆ 𝒥 (J),

  • (2)

    𝒥 (A ⋈^f J) = 𝒥 (A) ⋈^f 𝒥 (J),

  • (3)

    Jf(𝒥 (A)) ⊆ 𝒥 (J).

• (ii)

  If f(𝒥 (A))J ⊆ 𝒥 (J) (or equivalently Jf(𝒥 (A)) ⊆ 𝒥 (J)), then

A ⋈fJ/JA ⋈fJ=A/JA⋈fJ/JJ. \left( {A \;{\bowtie ^f}J} \right)/{\cal J}\left( {A \;{\bowtie ^f}J} \right) = A/{\cal J}\left( A \right) \;{\bowtie ^f}J/{\cal J}\left( J \right).

From now on a ring R = A ⊕_▷ J will be called a semi-direct regular ring, if it satisfies the equality: (4.1) JR=JA⊕⊳JJ {\cal J}\left( R \right) = {\cal J}\left( A \right)\;{ \oplus _ \triangleright }\;{\cal J}\left( J \right) or equivalently, if any of these inclusions hold: JAJ⊆JJ, JJA⊆JJ. {\cal J}\left( A \right)J \subseteq {\cal J}\left( J \right),\;\;J{\cal J}\left( A \right) \subseteq {\cal J}\left( J \right).

Analogously a ring A ⋈^f J will be called an amalgamated regular ring, if it satisfies the equality: (4.2) JA⋈fJ=JA⋈fJJ {\cal J}\left( {A \;{\bowtie ^f}J} \right) = {\cal J}\left( A \right) \;{\bowtie ^f}{\cal J}\left( J \right) or equivalently, if any of these inclusions are true: fJAJ⊆JJ, JfJA⊆JJ. f\left( {{\cal J}\left( A \right)} \right)J \subseteq {\cal J}\left( J \right),\;\;\;Jf\left( {{\cal J}\left( A \right)} \right) \subseteq {\cal J}\left( J \right).

According to (3.2) we immediately obtain the following corollary.

In this section we extend some results from [8, 15] to associative rings without unity and give corresponding results for semi-direct regular rings. Recall that B = f(A) + J.

Below, we characterize local semi-direct regular rings.

Next fact generalizes [8, Theorem 2.13] to associative rings satisfying (4.2). It can be proved analogously to Proposition 5.1 using Corollary 4.4.

Note that the condition (ii) in Corollary 5.2 can be repalced by A is a local ring and J⊆JB. A\;{\rm{is\; a\; local\; ring\; and}}\;J \subseteq {\cal J}\left( B \right).

Indeed, if J is 𝒥-radical, then 𝒥 (J) = J. However J ◁ B, so [18, Corollary 3.2.4] gives that 𝒥 (J) = J ∩ 𝒥 (B). This means J ⊆ 𝒥 (B).

Conversely, if J ⊆ 𝒥 (B), then 𝒥 (J) = J by the same corollary. Hence we have proved the following fact.

[25, Theorem 15] together with Proposition 5.1 give a characterization of semi-direct regular rings which are local and uniquely clean.

Corollaries 5.2, 5.3 and 5.4 together with (3.1) yield the following conclusion which extends [8, Corollary 2.16] to non-commutative rings satisfying (4.2).

Proposition 5.6 provides necessary and sufficient conditions for a ring that is a semi-direct sum of rings to be left T-nilpotent. In particular we obtain a characterization of left T-nilpotent amalgamated rings.

Now we are ready to present a characterization of semi-direct regular rings which are left Steinitz.

Below, we provide a characterization of left Steinitz amalgamated regular rings. The proof of this result is similar to the proof of Theorem 5.8 using Corollaries 5.2, 5.3, and Proposition 5.6. This fact extends the main result in [15] (cf. Theorem 2.1 (2)) to associative rings without unity satisfying (4.2).