This paper by reviewing the main examples of small to medium-scale retractable roof structures covering building courtyards is to present an innovative rigid retractable roof system employing spatial frames for a courtyard of an existing building in Tabriz Islamic Art University. The courtyard is currently used for temporary exhibitions and gatherings whenever permitted by environmental conditions. The proposed retractable roof will extend the application of the building throughout whole year period and also adds to its beauty and functionality. One of the main advantages of this design that makes it as a good alternative for this building is the way that the roof is retracted in different segments separately in a regulated deployment process and its potential in being used in different stages of the deployment process. The proposed roof consists of four retractable zones all covered with transparent rigid material and a fixed central part being inspired by the patterns of Iranian historic architecture. The retractable parts are placed at four corners and composed of rigid panels sliding across each other. An actuating force is applied to the first panel of each module and consequently makes the other panels move throughout the associated fixed track.

A module M is called epi-retractable if every submodule of M is a homomorphic image of M. Dually, a module M is called co-epi-retractable if it contains a copy of each of its factor modules. In special case, a ring R is called co-pli (respectively, copri) if RR (respectively, RR) is co-epi-retractable. It is proved that if R is a left principal right duo ring, then every left ideal of R is an epi-retractable R-module. A co-pli strongly prime ring R is a simple ring. A left self-injective co-pli ring R is left Noetherian if and only if R is a left perfect ring. It is shown that a cogenerator ring R is a pli ring if and only if it is a co-pri ring. Moreover, if R is a left perfect ring such that every projective R-module is co-epi-retractable, then R is a quasi-Frobenius ring.

We call a module MR essentially retractable if HomR (M,N)¹0 for all essential submodules N of M. For a right FBN ring R, it is shown that: (i) A nonzero module R M is retractable (in the sense that HomR (M,N) ¹0 for all nonzero N£MR) if and only if certain factor modules of M are essentially retractable nonsingular modules over R modulo their annihilators. (ii) A non-zero module MR is essentially retractable if and only if there exists a prime ideal PÎAss (MR) such that HomR (M,N)¹0. Over semiprime right nonsingular rings, a nonsingular essentially retractable module is precisely a module with nonzero dual. Moreover, over certain rings R, including right FBN rings, it is shown that a nonsingular module M with enough uniforms is essentially retractable if and only if there exist uniform retractable R-modules {U i}iÎI and R-homomorphisms M¾®+iÎI U i¾®M with ab;¹0.

Generalizing concepts “right Bezout ” and “principal right ideal ” of a ring R to modules, an R-module M is called n-epiretractable (resp. epi-retractable) if every n-generated submodule (resp. submodule) of MR is a homomorphic image of M. It is shown that if MR is finitely generated quasi-projective 1-epi-retractable, then EndR(M) is a right Bezout (resp. principal right ideal) ring if and only if MR is n-epi-retractable for all n³1 (resp. epiretractable). For a ring R and an infinite ordinal b³|R|, the R-module M = FÅN is epi-retractable where F is a free R-module with a basis set of cardinality b and N is a g-generated R-module with g£b. A ring R is quasi Frobenius if every injective R-module is epi-retractable. Injective modules in s[NR] are epi-retractable for every NÎs [MR] if and only if every non-zero factor ring of S is a quasi Frobenius ring where S is an endomorphism ring of a progenerator in s[MR].

Several characterizations of a ring R is given for which any non-zero finitely generated module M is retractable in the sense that HomR(M,N) is non-zero whenever N is a non-zero submodule of M. Such rings are called finite retractable. It is shown that any ring being Morita equivalent to a commutative ring is finite retractable. Also, if the commutative ring is semi-Artinian then any non-zero module is retractable. The class of finite retractable rings is shown to be closed under Morita equivalence and finite direct products. Moreover, for a finite retractable ring R which is a right order in a ring Q, it is shown that Q is also finite retractable.