Unknown Book Proceedings Series Volume 00, XXXX An Example of Osofsky and Essential Overrings Gary F. Birkenmeier, Jae Keol Park and S. Tariq Rizvi Dedicated to Professor Carl Faith and Professor Barbara L. Osofsky Abstract. Osofsky has shown that the right injective hull E(RR) of a ring R, in general, does not have a ring multiplication which extends its R-module „ « Z4 2Z4 multiplication by using the ring R = , where Z4 is the ring of 0 Z4 integers modulo 4. We explicitly characterize all right essential overrings of this ring R and investigate various properties as well as interrelationships between these right essential overrings. As a byproduct, we determine various ring hulls for the ring R (e.g., right FI-extending, right extending, right quasi- continuous, right continuous, and right self-injective ring hulls). Moreover, we find an intermediate R-module SR between RR and ER which has a compatible ring structure that is right self-injective and another compatible ring structure on SR which is even not right FI-extending. Generalizing Osofsky’s example, we provide a class of A-algebras R such that E(RR) has no compatible ring structure, where A is a commutative QF-ring. 1. Introduction Throughout this paper all rings are associative with unity and R denotes such a ring. All modules are unitary and we use MR to denote a right R-module. If NR is a submodule of MR, then NR is said to be essential (resp., dense, also called rational ) in MR if for any 0 6= x ∈ M, there exists r ∈ R such that 0 6= xr ∈ N (resp., for any x, y ∈ M with 0 6= x, there exists r ∈ R such that xr 6= 0, and yr ∈ N). Recall that a right ring of quotients T of R is an overring of R such that RR is dense in TR. The maximal right ring of quotients of R is denoted by Q(R), and the right injective hull of R is denoted by E(RR). We say that T is a right essential overring of a ring R if T is an overring of R such that RR is essential in TR. Note that, for an overring T of a ring R, if RR is 2000 Mathematics Subject Classification. Primary 16D50, 16L60 Key words and phrases. Essential extension, Essential overring, (FI-) extending, Kasch ring, Osofsky compatibility, QF-ring, Ring hull. The authors thank the referee for his/her comments and suggestions for the improvement of this paper. The second author was supported in part by Busan National University, 2008-2010. c XXXX American Mathematical Society 1 2 GARY F. BIRKENMEIER, JAE KEOL PARK AND S. TARIQ RIZVI dense in TR then RR is essential in TR while the converse is not true. Thus a right essential overring T of a ring R can be considered as a “generalized version” of a right ring of quotients of R. Definition 1.1. Let R be a ring, E(RR) the injective hull of RR, and TR an intermediate module between RR and E(RR). Then we say that a ring structure (T, +, •) on T is compatible if the ring multiplication • extends the R-module mul- tiplication of T over R. We notice that for an intermediate R-module TR between RR and E(RR), (T, +, •) is a compatible ring structure if and only if (T, +, •) is a right essential overring of R. It is well known that if the injective hull E(RR) is a rational extension of the right R-module RR, then it has a unique compatible ring structure [Lam, Theorem 13.11, p.367]. Also it is well known that a rational extension TR of RR has a unique compatible ring structure whenever such a ring structure on TR exists. In a general setting, Osofsky [O1, O2, and O3] investigated rings R such that E(RR) has a compatible ring structure, and she provided an example of a ring R such that no injective hull of RR has a compatible ring structure. Also Lang [Lang] showed that for a commutative Artinian ring R, E(RR) has a compatible ring structure if and only if R = E(RR). Embedding an arbitrary ring into a right self-injective ring has always been a challenge in view of Osofsky’s work, Faith [F, p.308], has described the solution provided by Menal and Vamos [MV] to the question of “embedding of any ring in an FP-injective ring” as the realization of “a three-decade old dream of Ring Theory”. Menal and Vamos [MV] also presented a right FP-injective ring R such that no right injective hull of RR has a compatible ring structure. In honor of Osofsky’s contributions to the study of injective hulls of rings, in [BPR3] a ring R is called right Osofsky compatible if an injective hull E(RR) of R has a compatible ring structure. Osofsky [O2] has shown that the right injective hull E(RR) of a ring R, in gen- eral, does not have a ring multiplication which extends its R-module multiplication Z4 2Z4 by using the ring R = , where Z4 is the ring of integers modulo 4. 0 Z4 In this paper, we explicitly characterize all right essential overrings of this ring R and investigate various properties as well as interrelationships between these right essential overrings. As a byproduct, we determine various ring hulls for this ring R (e.g., right FI-extending, right extending, right quasi-continuous, right continuous, and right self-injective ring hulls). In particular, although E(RR) has no compati- ble ring structure, there does exist a right self-injective right essential overring for R. This result also illustrates the general fact that if R is an arbitrary ring with right essential overrings S and V with S a subring of V and S is right self-injective, then S = V . So if R is right Osofsky compatible, then E(RR) hjas no proper sub- ring intermediate between R and E(RR) which is right self-injective. Generalizing Osofsky’s example, we provide a class of A-algebras R such that E(RR) has no compatible ring structure, where A is a commutative QF-ring. Let MR be a right R-module. Then MR is said to be (FI-) extending if every (fully invariant) submodule of MR is essential in a direct summand of MR (see [BMR], [DHSW], and [BPR1]). A ring R is called right (FI-) extending if RR is AN EXAMPLE OF OSOFSKY AND ESSENTIAL OVERRINGS 3 (FI-) extending. Right extending rings have also been called right CS rings [CH]. A right extending ring R is right (quasi-) continuous (if whenever AR and BR are direct summands of RR with A ∩ B = 0, then AR ⊕ BR is a direct summand of ∼ RR) if whenever XR = YR ≤ RR and XR is a direct summand of RR, then YR is a direct summand of RR. It is well known that injective ⇒ continuous ⇒ quasi-continuous ⇒ extending ⇒ FI-extending Reverse implications are not true in general. A ring R is called right Kasch [Lam, p.280] if every maximal right ideal of R has a nonzero left annihilator. If R is a right Kasch ring, then R = Q(R). ess For R-modules MR and NR, we use NR ≤ MR and NR ≤ MR to denote that NR is a submodule of MR and NR is an essential submodule of MR, respectively. We use J(−), Soc(−), and | | to denote the Jacobson radical of a ring, the right socle of a ring, and the cardinality of a set, respectively. 2. Osofsky’s Example and Nonisomorphic Essential Overrings Let A = Z4, the ring of integers modulo 4, and let A 2A R = , 0 A which is a subring of the 2 × 2 upper triangular matrix ring over A. It is shown in [O2] that no injective hull of RR has a compatible ring structure (i.e., the ring R is not right Osofsky compatible). Interestingly, Osofsky was able to obtain this result without describing the injective hull of RR explicitly. Note that Q(R) = R since R is a right Kasch ring. In this section, we determine all possible right essential overrings of the ring R and their interrelationships. Moreover, we investigate various properties of these right essential overrings. Also we explicitly provide various ring hulls of the ring R. For f ∈ Hom(2AA,AA) and x ∈ A, we let (f · x)(a) = f(xa) for all a ∈ A. Let A ⊕ Hom(2A ,A ) A E = A A , Hom(2AA,AA) A where the addition on E is componentwise and the R-module multiplication of E over R is given by a + f b x y ax + f · x ay + f(y) + bz = g c 0 z g · x g(y) + cz a + f b x y for ∈ E and ∈ R, where a, b, c, x, y, z ∈ A and f, g ∈ g c 0 z Hom (2AA,AA). Proposition 2.1. ([BPR3, Proposition 2.6]) E is an injective hull of RR. Put f0 ∈ Hom(2AA,AA) such that f0(2a) = 2a for a ∈ A. By [BPR3, Lemma 2.5], Hom(2AA,AA) = f0 · A. Thus if f ∈ Hom(2AA,AA), then f = f0 · s for some s ∈ A. Note that Hom(2AA,AA) = {0, f0}. Therefore all possible intermediate R-modules between RR and ER are: 4 GARY F. BIRKENMEIER, JAE KEOL PARK AND S. TARIQ RIZVI A ⊕ Hom (2A ,A ) 2A E, V = A A , Hom (2AA,AA) A A ⊕ Hom (2A ,A ) A AA Y = A A ,W = , 0 A Hom (2AA,AA) A A 2A S = , Hom (2AA,AA) A A ⊕ Hom (2A ,A ) 2A AA U = A A ,T = , and R.