In issue #2 of Green Lantern (Oct 2011), Sinestro gave Hal Jordan a ring that he said he made himself.
How can he make such a ring? When? Is it merely a construct?
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In the current iteration of the DCnU (new DC Universe recently revamped with 52 starting titles) it is too early to determine what, if any changes have occurred with the Green Lanterns, their limits, strengths or capabilities as determined by the new canon.
Under the older DCU canon the rings and lanterns of the Corps were:
Here is a website called the Book of Oa, that lists previous powers of the Power Rings of the Green Lantern Corps.
It has always been stated that a ring-bearer's performance with their ring was an expression of their willpower (or whatever attribute their ring functions on, with the appearance of other colored rings, some may utilize other emotional/mental triggers to perform their capabilities.)
With the Green rings, willpower and intellect were the most effective tools used to manipulate the Green energy. Famed Green Lanterns such as Sinestro or Kilowogg, were known to be able to push their rings far beyond what the average Green Lantern seemed capable of. This made them legendary among the Corps and the most difficult tasks were often reassigned to them.
Normally, the Green Lantern rings were not shown creating new things out of old things but it should be technically capable of doing so, if the person using the ring has the theoretical knowledge of how to a machine might perform the task or the fundamental knowledge to create the transformation directly using the ring.
Sinestro rebuilding a bridge that can sustain itself as if it were never broken either attests to a prodigious intellect capable of understanding the mechanics of the bridge and rebuilding the bridge at the molecular level, and using the Green Energy to hold everything in place until the bridge could be placed anew as if it had never experienced any trauma. Or he used another unknown means to restore the bridge to its former glory, perhaps by moving the bridge backward in time until it simply is restored to its previous temporal state. Given the power of the rings potential, either could be a satisfactory conclusion.
In the previous DCU, a Green Lantern ring was capable of producing a copy of itself that was a fully functioning copy. This might have been to create a spare device, since without a Lantern as a power source, it was merely a receiving platform. It might have been to create other rings to train potential Lanterns.
It is unclear if the rings created were also made of Oanite (implying the ability to transmit matter or whether they were created from the Green Energy itself, converting energy to matter, physically possible, if very difficult). It is unlikely to be a construct since if it was, it would be subject to disappear if the primary ring's power were suddenly removed or depleted.
As to whether he could control the ring after Hal had started using it, is a matter of debate. A sufficiently sophisticated will can take control of a ring even if another, less powerful will is using it. This is not something seen very often by Green Lanterns.
As to making permanent changes using the ring, we are not currently sure if this is a new angle taken by the writers to allow the Green Lanterns the opportunity to use the rings in a new and different fashion or as an expression of Sinestro of Korugar's superior use of the ring and how he was once called the Greatest of All Green Lanterns.
I'm assuming that this is an illustration of Sinestro's greater ability with the Ring.
The Ring has always been described as being able to create "anything the wielder can image".
Traditionally this has been Bulldozers, Giant Fists, etc. In this issue of Green Lantern Sinestro demonstrates that the bridge disaster can be completely reversed using ring power, not just the people being saved. It illustraters that the Earth GLs have been thinking too small with regards to their powers.
He also says that he has control over Hal's new ring, which suggests that it is certainly only a construct
(Most) Power Rings can replicate themselves, so that a Lantern in need of emergency assistance can temporarily "deputize" another person. The most famous example of this occurred in Blackest Night, when Ganthet triggered this process in seven rings, one of each color (even Orange, though that one didn't end entirely well). But it has happened a few other times in the series as well. We've never seen a Power Ring maintain more than one copy of itself at a time, and the copy is usually only temporary: even the deputies from Blackest Night were not permanent (Ganthet, the green deputy in that arc, later joined the GLC as a more permanent member, but he forged a "proper" ring for himself at that time).
What you're seeing in this case is just an unorthodox application of that process. Sinestro "deputized" Hal on a more permanent basis than usually happens, but Hal is nevertheless wearing a deputy ring.