Step-by-step analysis of the disaccharide structure:

1. Analyze Ring (a):

   • Ring Type: Ring (a) is a six-membered ring containing five carbon atoms and one oxygen atom. Such a six-membered ring structure for a monosaccharide is called a pyranose ring. (A five-membered ring is a furanose).
   • Glycosidic Linkage Configuration: The bond connecting the two monosaccharide units is called a glycosidic linkage. The nature of this linkage (\\{alpha\} or $\\{\beta\\}$) is determined by the configuration at the anomeric carbon of the first monosaccharide, which is C-1 of ring (a).
   • In the Haworth projection of a D-sugar, if the substituent on the anomeric carbon (C-1) is on the opposite side (trans) of the ring from the $-CH_2OH$ group (at C-5), the configuration is $\\{\alpha\\}$. If it's on the same side (cis), the configuration is $\\{\beta\\}$.
   • In ring (a), the $-CH_2OH$ group at C-5 is pointing upwards.
   • The glycosidic bond at the anomeric carbon (C-1) is pointing downwards.
   • Since the glycosidic bond and the $-CH_2OH$ group are on opposite sides (trans), the linkage is an $\\{\alpha\\}$-glycosidic link.
   • Therefore, ring (a) is a pyranose unit involved in an $\\{\alpha\\}$-glycosidic linkage.

2. Analyze Ring (b):

   • Ring Type: Ring (b) is also a six-membered ring containing one oxygen atom, so it is also a pyranose ring.
   • Anomeric Configuration: The anomeric carbon of ring (b) (C-1') is not involved in the glycosidic linkage; it has a free hydroxyl (-OH) group. We can determine its configuration. The $-CH_2OH$ group at C-5' is pointing upwards. The -OH group at the anomeric carbon C-1' is also pointing upwards. Since they are on the same side (cis), the configuration of this anomeric center is $\\{\beta\\}$. Thus, ring (b) is a $\\{\beta\\}$-glucopyranose unit. The entire molecule shown is $\\{\beta\\}$-Maltose.

3. Evaluate the Options:

   • A: Ring (a) is pyranonse with $\\{\alpha\\}$-glycosidic link.

     • As determined in step 1, ring (a) is a pyranose ring, and the glycosidic link it forms is $\\{\alpha\\}$. This statement is correct.

   • B: Ring (a) is furanonse with $\\{\alpha\\}$-glycosidic link.

     • Ring (a) is a pyranose, not a furanose. This statement is incorrect.

   • C: Ring (b) is furanonse with $\\{\alpha\\}$-glycosidic link.

     • Ring (b) is a pyranose, not a furanose. This statement is incorrect.

   • D: Ring (b) is pyranonse with $\\{\alpha\\}$-glycosidic link.

     • Ring (b) is a pyranose, which is correct. However, the glycosidic link is defined by ring (a). The anomeric center of ring (b) itself has a $\\{\beta\\}$-configuration. Therefore, this statement is incorrect.

Conclusion

The only correct statement is A.